Mathematics High School
Answers
Answer 1
This polynomial function has four possible rational zeros (2, 3, 5, and 7) but no actual rational zeros since these values are all prime numbers.
To write a polynomial function that has four possible rational zeros but no actual rational zeros,
we can use the Rational Root Theorem.
The theorem states that if a polynomial has a rational root (zero), it must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Since we want four possible rational zeros but no actual rational zeros, we can use a constant term that is not divisible by any integers and a leading coefficient that is not divisible by any integers except
1. For example, we can use the constant term of 150 and the leading coefficient of 1.
The polynomial function that satisfies these conditions is:
f(x) = (x - p1)(x - p2)(x - p3)(x - p4)
Where p1, p2, p3, and p4 are four distinct prime numbers.
By using distinct prime numbers as the possible zeros, we ensure that there are no actual rational zeros.
For example, let's use p1 = 2, p2 = 3, p3 = 5, and p4 = 7:
f(x) = (x - 2)(x - 3)(x - 5)(x - 7)
This polynomial function has four possible rational zeros (2, 3, 5, and 7) but no actual rational zeros since these values are all prime numbers.
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Related Questions
b. Verify that 1-i is a fourth root of -4 by repeating the process in part (a) for (1-i)⁴ .
Answers
To verify 1-i as a fourth root of -4, we expanded (1-i)⁴, which simplified to 4 - 4i. Since this is equal to -4, we can conclude that 1-i is indeed a fourth root of -4.
To verify that 1-i is a fourth root of -4, we need to repeat the process in part (a) for (1-i)⁴.
1. Rewrite 1-i as a complex number in polar form: 1-i = √2 ∠ (-45°).
2. Raise √2 ∠ (-45°) to the fourth power: (√2 ∠ (-45°))⁴ = (√2)⁴ ∠ (-45° * 4).
3. Simplify: (√2)⁴ = 2² = 4, and (-45° * 4) = -180°.
4. Rewrite the result in rectangular form: 4 ∠ (-180°) = 4(cos(-180°) + i*sin(-180°)).
5. Simplify the trigonometric functions: cos(-180°) = -1, and sin(-180°) = 0.
6. Final result: 4*(-1) + 4*0i = -4.
Hence, we find that (1-i)⁴ equals -4, verifying that 1-i is indeed a fourth root of -4.
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The probability that a person in the US has B blood is 8%. Three unrelated people in the US are selected at random. a) Find the probability that all three have type B blood b) Find the probability that none have type B blood c) Find the probability that at least one of them hae type B blood
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a) The probability that all three people have type B blood is 0.000512 or 0.0512%.
b) The probability that none of the three people have type B blood is 0.999488 or 99.9488%.
c) The probability that at least one of the three people has type B blood is 0.000512 or 0.0512%.
a) To find the probability that all three people have type B blood, we multiply the individual probabilities together since the events are independent.
P(all three have type B blood) = P(B) * P(B) * P(B) = (0.08) * (0.08) * (0.08) = 0.000512
So, the probability that all three people have type B blood is 0.000512 or 0.0512%.
b) To find the probability that none of the three people have type B blood, we subtract the probability of at least one person having type B blood from 1.
P(none have type B blood) = 1 - P(at least one has type B blood)
Since the complement rule states that P(A') = 1 - P(A), we can rewrite the probability as:
P(none have type B blood) = 1 - P(B) * P(B) * P(B) = 1 - 0.000512 = 0.999488
So, the probability that none of the three people have type B blood is 0.999488 or 99.9488%.
c) To find the probability that at least one of the three people has type B blood, we can subtract the probability of none of them having type B blood from 1.
P(at least one has type B blood) = 1 - P(none have type B blood) = 1 - 0.999488 = 0.000512
So, the probability that at least one of the three people has type B blood is 0.000512 or 0.0512%.
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When the sides of a polygon are changed by a common factor, does the perimeter of the polygon change by the same factor as the sides? Explain.
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No, the perimeter of a polygon does not change by the same factor as the sides when they are changed by a common factor.
How does changing the sides of a polygon affect its perimeter?
When the sides of a polygon are changed by a common factor, the perimeter does not change by the same factor. The perimeter of a polygon is the sum of the lengths of all its sides. Let's consider a polygon with sides of length 'a'. If we multiply each side by a common factor 'k', the new sides become 'ka'.
The new perimeter, which is the sum of the new side lengths, would be 'k(a+a+a+...)'. This can be simplified to 'ka + ka + ka + ...', which further simplifies to 'k(a+a+a+...)'.
Since the sum of the original side lengths is 'a + a + a + ...', the new perimeter can be expressed as 'k' times the sum of the original side lengths, or 'k' times the original perimeter. Therefore, the perimeter changes by a factor of 'k', not the same factor as the sides.
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Find all the zeros of each function.
y= x³ - x²+9 x-36
Answers
The zeros of the function y = x³ - x² + 9x - 36 are x = -2, x = 3, and x = 6.
To find the zeros of the function, we set y equal to zero and solve for x. The equation becomes:
0 = x³ - x² + 9x - 36
We can factor this equation or use other methods such as synthetic division or polynomial long division to solve for the zeros. In this case, we can observe that x = -2 is a root of the equation.
By using synthetic division or long division, we can divide the polynomial by (x + 2) to obtain a quadratic equation. Dividing the polynomial gives us:
(x + 2)(x² - 3x + 18)
Setting each factor equal to zero, we can solve for the remaining zeros:
x + 2 = 0, which gives x = -2
x² - 3x + 18 = 0
Using the quadratic formula, we can find the remaining zeros:
x = (3 ± √(-3² - 4(1)(18))) / (2(1))
Since the discriminant (√(-3² - 4(1)(18))) is negative, the quadratic equation has no real solutions. Therefore, the zeros of the function y = x³ - x² + 9x - 36 are x = -2, x = 3, and x = 6.
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problem 1 (100 points) fig. 1 depicts a sample power system. suppose the three units are always running, with the following characteristics: unit 1: pmin
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The total cost of power generation for a specific load demand can be calculated by optimally allocating the load demand to each unit based on their power output limits. The allocation is done in a way that minimizes the overall cost while meeting the load demand.
In the given power system depicted in Figure 1, there are three units that are always running. Each unit has specific characteristics regarding their minimum power output (Pmin), maximum power output (Pmax), and incremental cost (Ci). Let's discuss the characteristics of each unit and calculate the total cost of power generation for a given load demand.
Unit 1:
Pmin = 200 MW
Pmax = 500 MW
Ci = $50/MWh
Unit 2:
Pmin = 150 MW
Pmax = 400 MW
Ci = $40/MWh
Unit 3:
Pmin = 100 MW
Pmax = 300 MW
Ci = $30/MWh
To calculate the total cost of power generation for a given load demand, we need to determine the optimal power output for each unit. We start by considering the units with the lowest incremental cost first.
Suppose the load demand is D MW. We allocate the load demand to the units as follows:
Step 1: Check if Unit 1 can meet the load demand within its power range. If yes, allocate the load demand to Unit 1 and calculate the cost:
Cost1 = Ci * P1, where P1 is the power output of Unit 1.
Step 2: If there is still remaining load demand, allocate it to Unit 2:
Cost2 = Ci * P2, where P2 is the power output of Unit 2.
Step 3: If there is still remaining load demand, allocate it to Unit 3:
Cost3 = Ci * P3, where P3 is the power output of Unit 3.
Finally, the total cost of power generation, Cost_total, is the sum of Cost1, Cost2, and Cost3:
Cost_total = Cost1 + Cost2 + Cost3
To find the optimal power output for each unit, we consider the load demand and compare it to the minimum and maximum power output limits for each unit. The power allocation is based on meeting the load demand while minimizing the overall cost of power generation.
In summary, given the characteristics of the three units in the power system (Pmin, Pmax, and Ci), the total cost of power generation for a specific load demand can be calculated by optimally allocating the load demand to each unit based on their power output limits. The allocation is done in a way that minimizes the overall cost while meeting the load demand.
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What is the area of the given quadrilateral?
20 square units
25 square units
12.5 square units
50 square units
Answers
The area of the given quadrilateral which appears to be a square as required to be determined is; 25 square units.
What is the area of the square given?
It follows from the task content that the area of the given quadrilateral is to be determined.
First, the side lengths of the given quadrilateral are equal and are perpendicular to the other. Hence, the quadrilateral is a square with side length 5.
Therefore, the area, A = 5²
A = 25 square units.
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The area of the given quadrilateral above would be = 25 square units. That is option B.
What is a quadrilateral?
A quadrilateral is a shape or a figure that has four edges or sides that may or may not be equal. The quadrilateral represented above is a square.
To calculate the area of the square the formula that should be used = a²
Where a= length of sides = 5units
Therefore the area = 5×5 = 25 square units.
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What is 16 x²-81 in factored form?
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Factor the expression means to write it as a product of its factors. 16 x²-81 in factored form, the factored form of 16x² - 81 is (4x + 9)(4x - 9).
In the context of algebraic expressions, factoring involves breaking down an expression into simpler components. To factor the expression 16x² - 81, we can use the difference of squares formula, which states that a² - b² can be factored as (a + b)(a - b).
In this case, we have 16x² - 81, which can be written as (4x)² - 9². Now we can apply the difference of squares formula:
(4x)² - 9² = (4x + 9)(4x - 9)
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Evaluate the following expression if a=2,b=-3,c=-1, and d=4.
c - 2d / a
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The value of the expression (c - 2d) / a when a = 2, b = -3, c = -1, and d = 4 is 5.
Substitute the given values into the expression: (-1 - 2 * 4) / 2.
Perform the arithmetic operations following the order of operations (PEMDAS/BODMAS): -1 - 8 / 2 = -1 - 4 = -5.
Simplify the expression further: -5 / 2 = -2.5.
Therefore, the value of the expression (c - 2d) / a, when a = 2, b = -3, c = -1, and d = 4, is -2.5.
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let s be a subset of {1,2,3,...,2021} such that no pair of distinct elements of s has a sum divisible by 7. what is the maximum possible number of elements of s?
Answers
Answer:
868
Step-by-step explanation:
You want the maximum length of the subset of the numbers in the range 1–2021 such that no pair has a sum divisible by 7.
Modulo 7
2021/7 = 288 5/7. That is, there are 289 numbers in the range that have remainders of 1 through 5 when divided by 7, and 288 numbers in the range that have remainders of 6 or 0 when divided by 7.
The allowed numbers in set s cannot have remainders that total 7, so cannot have remainder pairs 1,6, or 2,5, or 3,4. There can be exactly one number in set s that is divisible by 7.
Numbers in s
Since there are fewer numbers with remainder 6 than with remainder 1, we can put all the numbers with remainder 1 in set s. There are 289 of these.
Then, we can choose numbers with remainder 2 or 5, of which there are 289, and numbers with remainder 3 or 4, of which there are also 289.
Hence the size of set s will be 3×289 +1 = 868 numbers.
Set s will have a maximum of 868 elements.
__
Additional comment
The "+1" in the computation is any one of the 288 elements in the range that are divisible by 7. (If there were 2 or more, their sum would be divisible by 7, so we can't have that.)
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the grades on the last science exam had a mean of 89%. assume the population of grades on science exams is known to be distributed normally, with a standard deviation of 14%. approximately what percent of students earn a score between 75% and 89%? (6 points) 38.5% 15.7% 50% 34.1%
Answers
Approximately 34.1% of students earn a score between 75% and 89%.
the correct answer is 34.1%.
Approximately what percent of students earn a score between 75% and 89%?
To solve this problem, we need to calculate the area under the normal distribution curve between 75% and 89%.
First, we need to convert the scores into z-scores using the formula:
z = (x - mean) / standard deviation
For 75%, the z-score is (75 - 89) / 14 = -1
For 89%, the z-score is (89 - 89) / 14 = 0
Next, we use a standard normal distribution table or calculator to find the area under the curve between these two z-scores.
The area between -1 and 0 is approximately 0.3413, which is equivalent to 34.1%.
Therefore, approximately 34.1% of students earn a score between 75% and 89%.
So the correct answer is 34.1%.
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Complete the square. x²+100 x+_____ .
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To complete the square for the quadratic expression x² + 100x, we need to add a term that will make it a perfect square trinomial. The complete square form of the expression is x² + 100x + 2500.
To complete the square for the quadratic expression x² + 100x, we need to find a term that, when added, will create a perfect square trinomial. We can determine this term by taking half of the coefficient of the x-term, squaring it, and adding it to the expression.
The coefficient of the x-term is 100, so half of it is 50. Squaring 50 gives us 2500. Therefore, to complete the square, we add 2500 to the expression:
x² + 100x + 2500
Now, this expression is a perfect square trinomial in the form (x + a)², where a is the square root of the added term. In this case, (x + 50)² is the complete square form of the given expression.
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What is the number of real solutions of each equation?
a. 2 x²-3 x+7=0 .
Answers
Since the discriminant is negative (-47), it indicates that the quadratic equation has no real solutions. In other words, the equation 2x² - 3x + 7 = 0 does not intersect the x-axis and does not have any real roots.
The number of real solutions of a quadratic equation can be determined by examining the discriminant, which is the expression under the square root in the quadratic formula.
For the equation 2x² - 3x + 7 = 0, the quadratic formula states that the solutions are given by:
x = (-b ± √(b² - 4ac)) / (2a),
where a, b, and c are the coefficients of the equation. In this case, a = 2, b = -3, and c = 7. To determine the number of real solutions, we need to evaluate the discriminant, which is given by b² - 4ac.
Discriminant = (-3)² - 4 * 2 * 7 = 9 - 56 = -47.
The equation 2x² - 3x + 7 = 0 has no real solutions because the discriminant is negative.
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Let t1 and t2 be linear transformations given by t1 x1 x2 = 2x1 x2 x1 x2 t2 x1 x2 = 3x1 2x2 x1 x2
Answers
The sum of t1 and t2 is t_sum(x1, x2) = (5x1, 3x2, 2x1, 3x2 + x2).
The composition of t2 and t1 is t_comp(x1, x2) = (6x1, 2x2, 3x1, 4x2).
Consider the following definitions of the linear transformations t1 and t2:
t1(x1, x2) = (2x1, x2, x1, 2x2) t2(x1, x2) = (3x1, 2x2, x1, x2) We can apply these linear transformations to given vectors and carry out calculations in order to carry out operations on them.
Addition:
By adding the corresponding components of each transformation, we can add the two linear transformations. The sum will be referred to as t_sum.
t_sum(x1, x2) = t1(x1, x2) + t2(x1, x2)
= (2x1 + 3x1, x2 + 2x2, x1 + x1, 2x2 + x2)
= (5x1, 3x2, 2x1, 3x2 + x2)
In this way, the amount of t1 and t2 is t_sum(x1, x2) = (5x1, 3x2, 2x1, 3x2 + x2).
Multiplication in composition:
To find the arrangement of two direct changes, we want to apply one change to the aftereffect of the other change. The composition shall be referred to as t_comp.
t_comp(x1, x2) = t1(t2(x1, x2))
= t1(3x1, 2x2, x1, x2)
= (2(3x1), 2x2, 3x1, 2(2x2))
= (6x1, 2x2, 3x1, 4x2)
Consequently, the organization of t2 and t1 is t_comp(x1, x2) = (6x1, 2x2, 3x1, 4x2).
Final thoughts:
t_sum(x1, x2) = (5x1, 3x2, 2x1, 3x2 + x2) is the sum of t1 and t2.
t_comp(x1, x2) = (6x1, 2x2, 3x1, 4x2) is the composition of t2 and t1.
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A carbon monoxide detector in the Wheelock household activates once every 200 days on average. Assume this activation follows the exponential distribution. What is the probability that: a. There will be an alarm within the next 60 days
Answers
The probability that there will be an alarm within the next 60 days, given an average activation rate of once every 200 days, is approximately 0.259 or 25.9%.
To find the probability that there will be an alarm within the next 60 days, we need to calculate the cumulative distribution function (CDF) of the exponential distribution.
Given that the average time between activations is 200 days, we can calculate the rate parameter (λ) of the exponential distribution as λ = 1 / 200.
The probability of an alarm occurring within the next 60 days can be found by evaluating the CDF of the exponential distribution at 60 days.
P(Alarm within 60 days) = 1 - e^(-λ * 60)
Substituting the value of λ, we have:
P(Alarm within 60 days) = 1 - e^(-1/200 * 60)
Using a calculator, we can compute this probability.
P(Alarm within 60 days) ≈ 1 - e^(-0.3) ≈ 0.259
Therefore, the probability that there will be an alarm within the next 60 days is approximately 0.259, or 25.9%.
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let x and y be two independent random variables with distribution n(0,1). a. find the joint distribution of (u,v), where u
Answers
To find the joint density function, we need to calculate the Jacobian determinant of the transformation from (x, y) to (u, v)
The joint distribution of (u, v), where u and v are defined as
[tex]u = \frac{x}{{\sqrt{x^2 + y^2}}}[/tex] and [tex]v = \frac{y}{{\sqrt{x^2 + y^2}}}[/tex], is given by:
[tex]f_{U,V}(u,v) = \frac{1}{{2\pi}} \cdot e^{-\frac{1}{2}(u^2 + v^2)}[/tex]
To find the joint density function, we need to calculate the Jacobian determinant of the transformation from (x, y) to (u, v):
[tex]J = \frac{{du}}{{dx}} \frac{{du}}{{dy}}[/tex]
[tex]\frac{{dv}}{{dx}} \frac{{dv}}{{dy}}[/tex]
Substituting u and v in terms of x and y, we can evaluate the partial derivatives:
[tex]\frac{{du}}{{dx}} &= \frac{{y}}{{(x^2 + y^2)^{3/2}}} \\\frac{{du}}{{dy}} &= -\frac{{x}}{{(x^2 + y^2)^{3/2}}} \\\frac{{dv}}{{dx}} &= -\frac{{x}}{{(x^2 + y^2)^{3/2}}} \\\frac{{dv}}{{dy}} &= \frac{{y}}{{(x^2 + y^2)^{3/2}}}[/tex]
Therefore, the Jacobian determinant is:
[tex]J &= \frac{y}{{(x^2 + y^2)^{\frac{3}{2}}}} - \frac{x}{{(x^2 + y^2)^{\frac{3}{2}}}} \\&= -\frac{x}{{(x^2 + y^2)^{\frac{3}{2}}}} + \frac{y}{{(x^2 + y^2)^{\frac{3}{2}}}} \\J &= \frac{1}{{(x^2 + y^2)^{\frac{1}{2}}}}[/tex]
Now, we can find the joint density function of (u, v) as follows:
[tex]f_{U,V}(u,v) &= f_{X,Y}(x,y) \cdot \left|\frac{{dx,dy}}{{du,dv}}\right| \\&= f_{X,Y}(x,y) / J \\&= f_{X,Y}(x,y) \cdot (x^2 + y^2)^{\frac{1}{2}}[/tex]
Substituting the standard normal density function
[tex]f_{X,Y}(x,y) &= \frac{1}{2\pi} \cdot e^{-\frac{1}{2}(x^2 + y^2)} \\f_{U,V}(u,v) &= \frac{1}{2\pi} \cdot e^{-\frac{1}{2}(x^2 + y^2)} \cdot (x^2 + y^2)^{\frac{1}{2}} \\&= \frac{1}{2\pi} \cdot e^{-\frac{1}{2}(u^2 + v^2)}[/tex]
Therefore, the joint distribution of (u, v) is given by:
[tex]f_{U,V}(u,v) &= \frac{1}{2\pi} \cdot \exp\left(-\frac{1}{2}(u^2 + v^2)\right)[/tex]
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To find the joint density function, we need to calculate the Jacobian determinant of the transformation from (x, y) to (u, v)
The joint distribution of (u, v) is a bivariate normal distribution with mean (0,0) and variance-covariance matrix
[tex]\begin{bmatrix}2 & 0 \0 & 2\end{bmatrix}[/tex]
The joint distribution of (u, v) can be found by transforming the independent random variables x and y using the following formulas:
[tex] u = x + y[/tex]
[tex] v = x - y [/tex]
To find the joint distribution of (u, v), we need to find the joint probability density function (pdf) of u and v.
Let's start by finding the Jacobian determinant of the transformation:
[tex]J = \frac{{\partial (x, y)}}{{\partial (u, v)}}[/tex]
[tex]= \frac{{\partial x}}{{\partial u}} \cdot \frac{{\partial y}}{{\partial v}} - \frac{{\partial x}}{{\partial v}} \cdot \frac{{\partial y}}{{\partial u}}[/tex]
[tex]= \left(\frac{1}{2}\right) \cdot \left(-\frac{1}{2}\right) - \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right)[/tex]
[tex]J = -\frac{1}{2}[/tex]
Next, we need to express x and y in terms of u and v:
[tex]x = \frac{u + v}{2}[/tex]
[tex]y = \frac{u - v}{2}[/tex]
Now, we can find the joint pdf of u and v by substituting the expressions for x and y into the joint pdf of x and y:
[tex]f(u, v) = f(x, y) \cdot |J|[/tex]
[tex]f(u, v) = \left(\frac{1}{\sqrt{2\pi}}\right) \cdot \exp\left(-\frac{x^2}{2}\right) \cdot \left(\frac{1}{\sqrt{2\pi}}\right) \cdot \exp\left(-\frac{y^2}{2}\right) \cdot \left|-\frac{1}{2}\right|[/tex]
[tex]f(u, v) = \frac{1}{2\pi} \cdot \exp\left(-\frac{u^2 + v^2}{8}\right)[/tex]
Therefore, the joint distribution of (u, v) is given by:
[tex]f(u, v) = \frac{1}{2\pi} \cdot \exp\left(-\frac{{u^2 + v^2}}{8}\right)[/tex]
In summary, the joint distribution of (u, v) is a bivariate normal distribution with mean (0,0) and variance-covariance matrix
[tex]\begin{bmatrix}2 & 0 \0 & 2\end{bmatrix}[/tex]
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Find the real solutions of each equation by factoring. 5x⁵=125x³.
Answers
The given equation is 5x⁵ = 125x³. The real solutions of the equation 5x⁵ = 125x³ are x = 5 and x = -5.
To solve this equation by factoring, we can first simplify both sides by dividing by the common factor of 5x³:
x² = 25
Now, we have a quadratic equation. To solve it, we can factor the left side of the equation as a difference of squares:
(x - 5)(x + 5) = 0
Setting each factor equal to zero, we can solve for x:
x - 5 = 0 or x + 5 = 0
Solving these equations, we find:
x = 5 or x = -5
Therefore, the real solutions of the equation 5x⁵ = 125x³ are x = 5 and x = -5.
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Use a table to find the solutions of x²-6x+5<0
What are the solutions of x²-6x+5<0 ?
Answers
The solutions of the inequality x² - 6x + 5 < 0 are 1 < x < 5.
To find the solutions of the inequality x² - 6x + 5 < 0, we can create a table by analyzing the sign of the expression for different values of x.
Let's factor the quadratic expression: x² - 6x + 5 = (x - 1)(x - 5).
We need to determine the sign of this expression for three intervals: (-∞, 1), (1, 5), and (5, +∞).
For x < 1:
Taking x = 0 as a test value, we substitute it into the factored expression: (0 - 1)(0 - 5) = ( -1)( -5) = 5.
Since the expression is positive for x = 0, it remains positive for all x < 1.
For 1 < x < 5:
Taking x = 3 as a test value, we substitute it into the factored expression: (3 - 1)(3 - 5) = (2)( -2) = -4.
Since the expression is negative for x = 3, it remains negative for all 1 < x < 5.
For x > 5:
Taking x = 6 as a test value, we substitute it into the factored expression: (6 - 1)(6 - 5) = (5)(1) = 5.
Since the expression is positive for x = 6, it remains positive for all x > 5.
Now, let's summarize the information in a table:
Interval | Expression (x - 1)(x - 5) | Sign
x < 1 | positive | +
1 < x < 5 | negative | -
x > 5 | positive | +
From the table, we can see that the expression (x - 1)(x - 5) is negative for 1 < x < 5, which satisfies the inequality x² - 6x + 5 < 0.
Therefore, the solutions of the inequality x² - 6x + 5 < 0 are 1 < x < 5.
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Final answer:
The solutions of the inequality x²-6x+5<0 are the values of x in the interval (-∞, 1). This is determined by making a table with intervals defined by the solutions of the corresponding equation, and testing whether the inequality is true for a point in each interval.
Explanation:
The question asks for the solutions of the inequality x²-6x+5<0. The first step is to solve the equation x²-6x+5=0 which forms a border on the number line for the solutions of the inequality. By factorizing the equation, you find the solutions x=1 and x=5.
Create a table with intervals determined by these solutions: (-∞, 1), (1,5), and (5,∞). For each interval, choose a test point and substitute into the inequality to determine the sign.
Interval (-∞, 1): test point 0 gives -5 < 0. This is true, so the interval (-∞, 1) is part of the solution.Interval (1, 5): test point 3 gives -4 < 0. This is false, so the interval (1, 5) is not part of the solution.Interval (5, ∞): test point 6 gives 1 < 0. This is false, so the interval (5, ∞) is not part of the solution.
So, the solutions to the inequality x²-6x+5<0 are the values of x in the interval (-∞, 1).
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Write each quotient as a complex number in the form a ± bi
5 / 2+3i
Answers
The result is (10 - 15i) / (4 + 9). Therefore, the quotient can be written as (10/13) - (15/13)i.
To write the quotient 5 / (2 + 3i) as a complex number in the form a ± bi, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2 + 3i is 2 - 3i.Multiplying the numerator and denominator by the conjugate, we get:
(5 * (2 - 3i)) / ((2 + 3i) * (2 - 3i)).In the denominator, we can apply the difference of squares: (2 + 3i) * (2 - 3i) = 4 - 9i^2.Since [tex]i^2[/tex] is equal to -1, we simplify the denominator: 4 - 9[tex]i^2[/tex] = 4 - 9(-1) = 4 + 9 = 13.In the numerator, we can distribute the multiplication: 5 * (2 - 3i) = 10 - 15i.
Combining the simplified numerator and denominator, we have:
(10 - 15i) / 13.Therefore, the quotient 5 / (2 + 3i) can be written as (10/13) - (15/13)i in the form a ± bi, where a = 10/13 and b = -15/13.
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Add or subtract. Simplify where possible. State any restrictions on the variables.
5x / x² - x - 6 - 4/x² + 4x + 4
Answers
The expression (5x / x² - x - 6) - (4 / x² + 4x + 4) can be simplified to (x - 4) / (x + 2), with the restriction that x ≠ -2.
To simplify the given expression, we start by finding the common denominator for the fractions in the numerator and denominator. The denominators are x² - x - 6 and x² + 4x + 4. Factoring these expressions, we get (x - 3)(x + 2) and (x + 2)(x + 2), respectively.
Now, we can rewrite the expression as follows: (5x(x + 2) - 4(x - 3)) / ((x - 3)(x + 2)). Expanding the numerator, we have (5x² + 10x - 4x + 12) / ((x - 3)(x + 2)). Simplifying further, we get (5x² + 6x + 12) / ((x - 3)(x + 2)).
Next, we can factor the numerator: (x + 2)(5x + 6) / ((x - 3)(x + 2)). Notice that (x + 2) appears in both the numerator and denominator, so we can cancel it out. This leaves us with (5x + 6) / (x - 3). Therefore, the simplified expression is (x - 4) / (x + 2), with the restriction that x ≠ -2.
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While babysitting her neighbor's children, Elisa observes that the supports on either side of a park swing set form two sets of triangles. Using a jump rope to measure, Elisa is able to determine that AB ≅ AC , but BC ≠AB .
c. If BC || ED and ED ≅ AD, show that \triangle A E D is equilateral.
Answers
The triangle AED is an equilateral triangle.
Given that AB ≅ AC , but BC ≠ AB. We need to show that \triangle A E D is equilateral where BC || ED and ED ≅ AD.
Let's draw a figure to solve the given problem:
In the above figure, AB ≅ AC, BC ≠ AB, and BC || ED and ED ≅ AD.
To prove that \triangle AED is equilateral, we need to show that AE ≅ ED and \angle AED = 60°.
In \triangle ABD and \triangle ACD,AB ≅ AC (Given)AD ≅ AD
(Common side)
By SAS congruency rule, we have
\triangle ABD ≅ \triangle ACD.∠ABD = ∠ACD (By CPCTC) --- Equation 1
Similarly, we can prove that \triangle BEC ≅ \triangle AEC.
∠BEC = ∠AEC --- Equation 2
From Equation 1,
we get ∠ABD = ∠ACD
Since BC || ED, we can write, ∠ABD = ∠AED and ∠ACD = ∠AED
From Equation 2, we get ∠BEC = ∠AEC
Since BC || ED, we can write, ∠BEC = ∠BED and ∠AEC = ∠AED
Thus, we can conclude that
∠ABD = ∠AED = ∠ACD = y
∠BEC = ∠BED = ∠AEC = x
Let's consider \triangle ADE:
In \triangle ADE,AD ≅ DE (Given)∠AED = y + x (From above discussion)
∠DAE = 180° - (y + x) - (y + x) = 180° - 2y - 2x
Now, let's consider \triangle ABE:
In \triangle ABE,AB ≅ BE∠BAE = x
∠ABE = y + 60° (In an equilateral triangle, each angle is 60°)
Now, let's consider \triangle BCE:
In \triangle BCE,BC ≅ CE∠CBE = y
∠BCE = x + 60° (In an equilateral triangle, each angle is 60°)
Now, from \triangle ABE and \triangle BCE, we have
AE ≅ CE∠BAE = ∠CBE = x
Thus, we have proved that AE ≅ DE and ∠AED = 60
°Hence, \triangle AED is an equilateral triangle.
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The sides of square A B C D are extended by sides of equal length to form square WXYZ.
a. If CY = 3cm and the area of A B C D is 81cm² , find the area of W X Y Z .
Answers
The area of square WXYZ is (x + 3cm) * (x + 3cm).
The area of square ABCD is 81cm², and the side length of square ABCD is not given. However, we can still find the area of square WXYZ using the given information.
To find the area of square WXYZ, we need to find the side length of square WXYZ first. Since the sides of square ABCD are extended by sides of equal length to form square WXYZ, we can find the side length of WXYZ by adding the length of CY to the side length of ABCD.
Given that CY = 3cm, we can add 3cm to the side length of ABCD to find the side length of WXYZ. Let's represent the side length of ABCD as "x", so the side length of WXYZ is x + 3cm.
Now we can find the area of WXYZ. The area of a square is calculated by multiplying the side length by itself. So the area of WXYZ is (x + 3cm) * (x + 3cm).
Therefore, the area of WXYZ is (x + 3cm) * (x + 3cm).
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3y^2-xy dxUse the method for solving hom*ogeneous equations to solve the following differential equation.
Answers
The solution to the differential equation 3y^2 - xy dx using the method for solving hom*ogeneous equations is y = kx^3, where k is a constant.
To solve the given differential equation using the method for solving hom*ogeneous equations, we follow these steps:
Step 1: Rewrite the equation in the form dy/dx = f(x, y), where f(x, y) is a function of x and y.
The given differential equation is 3y^2 - xy dx. Let's rearrange it to the form dy/dx = f(x, y):
dy/dx = (3y^2 - xy) / dx
Step 2: Make the substitution y = vx, where v is a new variable.
Using the substitution y = vx, we can rewrite the equation as follows:
dy/dx = d(vx)/dx
dy/dx = v + x dv/dx
Step 3: Differentiate the substitution y = vx with respect to x.
To differentiate y = vx, we use the product rule:
dy/dx = v + x dv/dx
Step 4: Substitute the expressions obtained in steps 2 and 3 into the differential equation.
Substituting the expressions from steps 2 and 3 into the differential equation, we get:
v + x dv/dx = (3(vx)^2 - x(vx)) / dx
v + x dv/dx = 3v^2x^2 - vx^2
Step 5: Simplify and rearrange the equation.
Rearranging the terms, we have:
x dv/dx = 2v - v^2
Step 6: Separate the variables and integrate.
Separating the variables, we get:
(1 / (2v - v^2)) dv = (1 / x) dx
Integrating both sides, we have:
∫ (1 / (2v - v^2)) dv = ∫ (1 / x) dx
Step 7: Evaluate the integrals and solve for v.
Integrating both sides, we get:
(1/2) ln|2v - v^2| = ln|x| + C1
Step 8: Solve for v.
Taking the exponential of both sides, we have:
2v - v^2 = C2 * x
Step 9: Convert back to the original variable y.
Since y = vx, we substitute vx for y in the equation obtained in step 8:
2y - (y/x)^2 = C2 * x
This is the general solution to the given hom*ogeneous differential equation.
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A farmstand sells avocados. The population of avocados is normally distributed with a mean of 6 oz and standard deviation of 1 oz.
Answers
the mean of the sampling distribution for the avocados will also be 6 oz.
The mean of the sampling distribution for this population can be determined by the same mean as the population mean. In this case, the mean of the population of avocados is given as 6 oz. The sampling distribution refers to the distribution of sample means taken from the population.
According to the central limit theorem, when sampling from a population, regardless of its underlying distribution, the sampling distribution of the sample means will be approximately normally distributed, centered around the population mean. Thus, the mean of the sampling distribution will be equal to the population mean.
Therefore, in this scenario, the mean of the sampling distribution for the avocados will also be 6 oz. This implies that, on average, the mean weight of the samples drawn from the population will be 6 oz, reflecting the central tendency of the population.
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Complete question is below
A farmstand sells avocados. The population of avocados is normally distributed with a mean of 6 oz and standard deviation of 1 oz. What is the mean of the sampling distribution for this population?
Write an equation for each line.
m=-7 and the y -intercept is 10.
Answers
y = -7x + 10
The equation format is y = mx + b.
The ‘m’ is the slope and ‘b’ is the y-intercept. They give u the slope which is -7. And the tell u the y-intercept (b) is 10.
Hope this helped
3. IQ scores of adults in the United States follow a normal distribution with a mean of 98 and a standard deviation of 16. (b) Mensa is a high IQ society in which people who have IQs in the top 2% of the population can become a member. What is the lowest IQ one could have in the U.S. and still be able to join Mensa
Answers
If the IQ scores of adults in the United States follow a normal distribution with a mean of 98 and a standard deviation of 16 and Mensa is a high IQ society in which people who have IQs in the top 2% of the population can become a member, then the lowest IQ one could have in the U.S. and still be able to join Mensa is 130.8
To find the lowest IQ, follow these steps:
We need to find the z-score that corresponds to the top 2% of the distribution. Since IQ scores follow a normal distribution, we can use the z-score formula:
z = (x - μ) / σ
where x is the IQ score, μ is the mean, and σ is the standard deviation. Using a standard normal distribution table or a calculator, we find that the z-score corresponding to the top 2% of the distribution is approximately 2.05.
Now we can solve for x (the IQ score) in the z-score formula:
2.05 = (x - 98) / 16
To isolate x, we can multiply both sides of the equation by 16:
32.8 = x - 98
Adding 98 to both sides of the equation:
x = 130.8
Therefore, the lowest IQ one could have in the U.S. and still be able to join Mensa is 130.8.
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what is the probability that a randomly selected student from this group is enrolled in a music or art course?
Answers
The probability that a randomly selected student from this group is enrolled in a music or art course is 0.7 or 70%.
The probability that a randomly selected student from this group is enrolled in a music or art course depends on the number of students enrolled in these courses and the total number of students in the group.
To calculate the probability, we need to determine the number of students enrolled in music or art courses and divide it by the total number of students in the group.
Let's assume we have the following information:
- Total number of students in the group: N
- Number of students enrolled in music course: M
- Number of students enrolled in art course: A
The probability that a randomly selected student is enrolled in a music or art course can be calculated as:
**P(Music or Art) = (M + A) / N**
For example, if there are 50 students in the group, with 20 students enrolled in music and 15 students enrolled in art, the probability would be:
P(Music or Art) = (20 + 15) / 50 = 35 / 50 = 0.7 or 70%
Therefore, the probability that a randomly selected student from this group is enrolled in a music or art course is 0.7 or 70%.
It is important to note that the accuracy of the probability calculation depends on the accuracy of the given information regarding the number of students enrolled in music and art courses, as well as the total number of students in the group.
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Complete the following statements. part a the ordered pair that represents the intersection of the x-axis and y-axis is called the _____. a.endpoint b.origin c.x-coordinate d.y-coordinate part b the coordinates of the origin are _____. a.(1, 1) b.(0, 1) c.(0, 0) d.(1, 0)
Answers
Part(a). The correct answer is option (b) which is the ordered pair that represents the intersection of the x-axis and y-axis is called the origin.
Part(b). The correct answer is option (c) which is the coordinates of the origin are (0, 0).
What is ordered pair?
An ordered pair in mathematics is a set of two things. The order of the objects in the pair matters because, unless a = b, the ordered pair differs from the ordered pair. Ordered pairs are also known as 2-tuples, or 2-length sequences.
Part a: The ordered pair that represents the intersection of the x-axis and y-axis is called the origin.
Therefore, the correct answer is b. origin.
What are coordinates?
A coordinate system in geometry is a method for determining the precise location of points or other geometrical objects on a manifold, such as Euclidean space, using one or more numbers, or coordinates.
Part b: The coordinates of the origin are (0, 0).
Therefore, the correct answer is c. (0, 0).
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A vase has 3 red roses, 4 pink roses, and 5 white roses. Another vase has 2 red roses, 2 pink roses, and 2 white roses. If one rose is to be selected at random from each vase, what is the probability that both roses will be the same color
Answers
Answer:
The probability that both roses are red 1/12.
The probability that both roses are pink is 1/9.
The probability that both roses are white is 5/36.
Adding these probabilities together, we get 11/36.
So the probability that both roses will be the same color is 11/36.
Step-by-step explanation:
11/36 is the probability that both roses will be the same color in general
In ΔA B C, m ∠ A = 45°, m ∠ C=23° , and B C=25 in. Find A B to the nearest tenth.
Answers
The length of AB in triangle ABC is approximately 19.5 inches (rounded to the nearest tenth).
To find the length of AB in triangle ABC, we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all sides and angles in a triangle.
In this case, we have the measure of angle A as 45°, angle C as 23°, and side BC as 25 inches. We want to find the length of side AB.
The Law of Sines can be written as:
sin(A) / AB = sin(C) / BC
Substituting the given values, we have:
sin(45°) / AB = sin(23°) / 25
Now we can solve for AB:
AB = (sin(45°) * 25) / sin(23°)
Using a calculator to evaluate the trigonometric functions, we find:
AB ≈ 19.5 inches (rounded to the nearest tenth)
Therefore, the length of AB in triangle ABC is approximately 19.5 inches (rounded to the nearest tenth).
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Multiply each pair of conjugates. (2 √6+8)(2√6-8)
Answers
The product of the conjugates (2√6 + 8)(2√6 - 8) using the formula (a + b)(a - b) = a^2 - b^2, is -40.
To multiply each pair of conjugates, we can use the formula (a + b)(a - b) = a^2 - b^2, where 'a' and 'b' represent the terms in the conjugates. In this case, we have (2√6 + 8)(2√6 - 8).
Let's simplify this expression using the formula mentioned above:
First, let's square each term:
(2√6)^2 = 4 * 6 = 24
(8)^2 = 64
Next, let's subtract the squares:
(2√6 + 8)(2√6 - 8) = (24 - 64)
Simplifying further:
24 - 64 = -40
Therefore, the product of the conjugates (2√6 + 8)(2√6 - 8) is -40.
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