A minimum sample size of 1399 is required assuming a prior knowledge of the sample proportion of 11%, and a minimum sample size of 960 is required if no prior knowledge of the sample proportion is assumed.
To obtain an estimate of the percentage of U.S. adults with pinworm with an error of no more than 1.5 percentage points at a 95% confidence level, a minimum sample size of 1399 is required, assuming prior knowledge of the sample proportion of 11%.
However, if no prior knowledge of the sample proportion is assumed, a minimum sample size of 960 is required.
To calculate the minimum sample size, the formula n = (Z² * p * (1-p)) / E² is used, where n is the sample size, Z is the Z-value for the desired confidence level (1.96 for 95% confidence), p is the prior sample proportion, and E is the desired margin of error.
Therefore, assuming a prior knowledge of the sample proportion of 11%, a sample size of 1399 was needed. If no prior knowledge was assumed, a minimum sample size of 960 was required.
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Geometry: Find the unknown lengths of these isosceles right triangles (ASAP!!)
22=4 square root 2
Step-by-step explanation:
Using Pythagorean Theory
Surface area of the prism
Answer: 450
Step-by-step explanation:
9x10x5
During gym class, there were 7 teams in the four-square match. Each team was made of 5 third-graders.
Which shows how many students were in the four-square match in all?
1. 5 times 5
2. 5 minus 7
3. 7 times 5
4. 7 plus 5
7 times 5 shows the number of students in the four-square match. The solution has been obtained by using arithmetic operations.
What are arithmetic operations?
It is stated that the four fundamental operations, often known as "arithmetic operations," can explain all real numbers. Quotient, product, sum, and difference are the next four mathematical operations after division, multiplication, addition, and subtraction.
We are given that there were 7 teams in the four-square match and each team was made of 5 third-graders.
Now, in order to find the total students, we will use the multiplication operation.
From this, we get 7 times 5 which represents the total students.
Hence, 7 times 5 shows the number of students in the four-square match.
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The sum of two numbers is 58 and the difference is 20. What are the numbers?
Larger number:
Smaller number:
Answer:
Larger number: 39
Smaller number: 19
Step-by-step explanation:
Write two equations and solve the system of equations.
One number can be x. The other number can be y.
"sum" means adding.
x + y = 58
"difference" means subtracting.
x - y = 20
Take both equations, stack them and add them together from the top down.
x + y = 58
x - y = 20
_________
2x = 78
Divide by 2
x = 39
Use either original equation to find the other number.
x + y = 58
39 + y = 58
subtract 39
y = 19
The bigger number is 39 and the smaller number is 19.
the area of rectangular surface of a triangular prism having base sides 9 CM 10 cm and 17 cm is 864 CM square calculate the height of the prism
Answer:
The formula for the volume of a triangular prism is given by:
V = (1/2)bhL
where b is the base length of the triangle, h is the height of the triangle, and L is the length of the prism (i.e., the distance between the two triangular faces).
We are given the area of the rectangular surface of the prism, which is the product of the lengths of two adjacent sides of the rectangle, and we know that this area is equal to 864 cm^2. Therefore, we can use the Pythagorean theorem to find the length of the third side of the rectangle, which is also the base length of the triangular face:
17^2 = 9^2 + 10^2
289 = 81 + 100
289 = 181
So, the base length of the triangular face is 8 cm. Now we can plug in the known values into the formula for the volume of the prism:
864 = (1/2)(8 cm)(h)(L)
We are not given the length of the prism, L, but we know that it is equal to the base length of the triangle, which is 8 cm. Therefore, we can simplify the equation to:
864 = 32h
Solving for h, we get:
h = 864/32
h = 27
So the height of the prism is 27 cm.
5.8 Khomo and Peter bought a house for R575 000 as an investment. Khomo payed R245 000 and Peter payed the rest. They sold the house 5 years later and made a profit of R234 500.if they share the profit in the same ratio as their respective investments,how much profit will peter recieve
Answer:
133,380
Step-by-step explanation:
Cost of house 575,000
Khomo paid 245,000
Peter paid 575,000 - 245,000 = 330,000
Ratio of Khomo to Peter = 245,000 : 330,000 = 49:66
Profit made = 234,500
Khomos share is 49/ 115 x 234,00 = 0.43 x 234,000 = 100,620
Peters share is 66/115 x 234,000 = =0.57 x 234,000 = 133,380
I Need help with this
First person to answer gets brainilest.Do what the picture says!!!!
Answer:
please mark as brainliest
Answer:Good question
Step-by-step explanation:
Find a formula for the exponential function passing through the points (5, 32567 ) and (4, 16
189). Use exact values, not decimals. f(x)=
As a result, the exponential function travelling between the points (5, 32567) and (4, 16189) has the following formula: function [tex]f(x) = 14000 * (1.007044628)^x[/tex]
what is function?Mathematicians examine numbers with their modifications, equations and associated structures, forms and their locations, and feasible positions for these things. The term "function" indicates the connection between a collection of inputs, each of which has a corresponding output. A function is a connection of inputs and outputs where each supply leads to a specific, identifiable outcome. Each function is assigned a domain, a codomain, or a scope. The letter f is widely used to denote functions (x). The symbol for entry is an x. The four primary types of accessible capabilities are on functions, yet another skills, so multiple capabilities, in capabilities, and on operations.
f(x) = a * bx, where a and b are unknown constants. We have two places through which the function passes, which implies we have two equations:
[tex]f(5) = a * b^5 = 32567 f(4) = a * b^4 = 16189[/tex]
When we divide these two equations, we get:
[tex]b^5 / b^4 = 32567 / 16189 b = (32567 / 16189)^(1/1) = 1.007044628[/tex]
We may solve for a by substituting this value of b in any of the equations:
[tex]a = 16189 / b^4 = 16189 / (1.007044628)^4 = 14000[/tex]
As a result, the exponential function travelling between the points (5, 32567) and (4, 16189) has the following formula:
[tex]f(x) = 14000 * (1.007044628)^x[/tex]
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What is the equation for part a?
The rectangle shown has a perimeter of 52 inches.
a. Write an equation describing the possible values of x and y.
Using the perimeter formula of a rectangle we can write an equation describing the values of x and y is as follows:
2y + 2x = 52 inches
Define perimeter?The perimeter of a rectangle is the total length of its edges. It is expressed in linear units like centimetres, inches, and the like and can be taken to mean the total measurement of the rectangle's length and width. For instance, you may rapidly calculate how much ribbon you'd need if you needed to decorate the edge of your rectangular notebook. Similar to how calculating the perimeter will help you determine how much wire you'll need to erect a fence around your garden.
The formula P = 2(l + w), where l is the length and w is the width, determines the perimeter of a rectangle. Since we are aware that this rectangle's perimeter is 52 inches, we can write the following equation: 2l + 2w = 52 inches.
We can express the equation in terms of x and y to discover the potential values for x and y. We may substitute the width (x) and the length (y) into the equation to get 52 inches: 2y + 2x.
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A runner sprinted for 447 feet . How many yards is this
Answer: 447/3
Feet to yards is every three feet is one yard
Step-by-step explanation:
447 divided by 3 equals 149 yards
Explain. What should the format of a problem be if using the Factoring Method? Can this method be applied for any quadratic problems? What are the limitations? Why do we make each factor equal to zero?
The factoring method is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. To use the factoring method, we need to find two numbers that multiply to give c and add to give b. We then use these two numbers to factor the quadratic expression into two binomials, set each binomial equal to zero, and solve for x.
The format of a problem that can be solved using the factoring method should be in the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. The quadratic expression should be factorable into two binomials.
The factoring method can be applied to any quadratic problem that is in the correct format and is factorable. However, not all quadratic equations can be factored. In cases where the quadratic expression cannot be factored, we need to use other methods such as the quadratic formula or completing the square.
The main limitation of the factoring method is that it only works for quadratic equations that can be factored into two binomials. If the quadratic equation cannot be factored or if it has complex roots, then the factoring method cannot be used to solve the problem.
We make each factor equal to zero because if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x to find the roots of the quadratic equation.
IF YOU GET THIS ONE QUESTION RIGHT, I'LL GIVE BRAINLIEST
Solve this equation.
√(x-10)^2=x-10
Use this tax table to find how much tax you need to pay on a taxable income of $25,000.
If taxable income is over-- But not over-- The tax is:
$0 $7,825 10 percent of the amount over $0
$7,825 $31,850 $782. 50 plus 15 percent of the amount over 7,825
$31,850 $77,100 $4,386. 25 plus 25 percent of the amount over 31,850
$77,100 $160,850 $15,698. 75 plus 28 percent of the amount over 77,100
$160,850 $349,700 $39,148. 75 plus 33 percent of the amount over 160,850
$349,700 no limit $101,469. 25 plus 35 percent of the amount over 349,700
On a taxable income of $25,000, you must pay $3,358.75 in taxes.
To find how much tax you need to pay on a taxable income of $25,000, you will need to use the tax table given.
Since $25,000 falls in the third row of the table, which is for taxable income over $7,825 but not over $31,850, we will need to use the formula for this row:
Tax = $782.50 + 15% of the amount over $7,825
To calculate the tax, we need to first find the amount over $7,825, which is:
25,000 - 7,825 = 17,175
Now we can substitute this into the formula to get:
Tax = 782.50 + 0.15 x 17,175
Tax = 782.50 + 2,576.25
Tax = 3,358.75
Therefore, the amount of tax you need to pay on a taxable income of $25,000 is $3,358.75.
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If the total surface area of a cube is 302.46 in², which best describes the length of an edge of the cube?
The length of the edge of the cube with the given total surface area is 7.1 inches.
What is total surface area?The region that includes the base(s) and the curved portion is referred to as the total surface area. It is the overall area that the object's surface occupies. The total area of a form with a curved base and surface is equal to the sum of the two areas.
Let us suppose the length of an edge of the cube = l.
The total surface are of the cube is given as:
TSA = 6l²
Substituting TSA = 302.46 we have:
302.46 = 6l²
l² = 50.41
l = 7.1 in.
Hence, the length of the edge of the cube with the given total surface area is 7.1 inches.
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Because of your amazing work maximizing the revenue for the math club, the basketball team wants you to help them. They want to know how much admission they should charge for their games to make the most money. The team manager tells you that when they charged $5 per ticket, they sold 600 tickets, but when they tried to charge $20 per ticket, they sold no tickets.
1. You can assume the relationship between the price of the ticket and the number of tickets sold is linear. Write a linear function “n” that models the number of tickets sold “n(p),” as a function of the price of a ticket, “p.”
2. To calculate revenue, you multiply the price of the ticket by the number of tickets sold. Write a function “r” that models the revenue generated by selling tickets, “r(p),” as a function of the price of the ticket, “p.”
3. Use your function “r” from question 2 to recommend a ticket price to the basketball team. Support your recommendation by explaining how much revenue they could generate from your ticket price versus a higher or lower price.
1. Linear function: n(p) = -40p + 800
2. r(p) = P. n(p)
3. Total revenue = $4000
What is the slope of a linear function?There are numerous formulas for deriving a line's equation using linear functions, depending on the available data.
With the exception of vertical lines, which are not functions, any line may have its equation found using a formula for a linear function.
In the question,
We can assume, we have two points:
(5, 600) and (20,0)
The slope:
0-600/20-5
= -40
So, n(p) = -40 (p-20)
n(p) = -40p +800
Now solving,
r(p) = P. n(p)
= -40p² + 800p
Now to calculate the revenue,
r'(p) = -2 × 40p + 800
= -80p + 800
Order r'(p) = 0
⇒ P = 10
The highest price is: $10.
Revenue = -40 × ($10) ² + 800 × $10 = $4000
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A quantity m varies jointly with p and r. When p is 2 and r is 4, m is 1. What is the constant of variation?
One-eighth
One-half
2
8
The constant of variation is [tex]\frac{1}{8}[/tex].
What is variation?The discrepancy among an ideal and real scenario is referred to as the Law of Variation. The most frequent manifestations of variation or variability are changes in the data, anticipated results, or minor changes in the quality of the output.
The types of variation are given below-
Where a variable is a fixed multiple of another, this is known as direct variation. For instance, my income varies directly (or proportionally) with the amount of hours I put in.When one of the variables rises, the other one falls—this is known as inverse and indirect variation (product is constant). For instance, the length of time an air conditioner is operating has an indirect (equal or inverse) impact mostly on temperature in my home.Joint variation occurs when at least two factors are directly associated. For instance, a triangle's base and height are both related to the triangle's area.According to the question;
A quantity m varies jointly with p and r which shows that p and r in joint variation with m.
m ---> pr
m = kpr
where, k is the constant of variation.
m = 1, p = 2, r = 4.
Substitute the values in the relation,
1 = k(2)(4)
1 = 8k
k = 1/8
Therefore, the constant of variation is [tex]\frac{1}{8}[/tex].
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The constant of variation is 1/8, which is option A.
What is variation?
The way in which a quantity changes or varies with respect to another quantity. There are two types of variation: direct variation and inverse variation. Direct variation occurs when two quantities change in the same direction. In other words, if one quantity increases, the other quantity also increases.
y = kx, where y and x are the two quantities, and k is a constant of proportionality.
Inverse variation occurs when two quantities change in opposite directions. In other words, if one quantity increases, the other quantity decreases.
y = k/x, where y and x are the two quantities, and k is a constant of proportionality.
If m varies jointly with p and r, we can write the relationship as:
m = k * p * r
where k is the constant of variation.
We are given that when p is 2 and r is 4, m is 1. Substituting these values into the equation, we get:
1 = k * 2 * 4
Simplifying this equation, we get:
k = 1 / (2 * 4)
k = 1/8
Therefore, the constant of variation is 1/8, which is option A.
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Jessica earns 52 dollars per week working part- time at a book store she makes one dollar more for each book sells the amount,A(in dollars), that Jessica earns in a week if she sells b books is given by the following
The amount that Jessica will earns if she sells 34 books is 86 dollars.
How much will Jessica earns?An earning refers to the income that a person get from wages and salaries, security and other government benefits, dividends and interest, business ownership, other sources etc.
Our Equation to solve for earnings is:
A = 52 + (1)B
A = 52 + B
A(34) = 52 +34
A(34) = $86
Therefore, the total earned in one week if she sold 34 books is $86.
Missing words' Write an equation relating A to B. Then use this equation to find the amount of money Lena earns if she sells 34 books. Equation: Amount Jessica earns is she sells 34 books: ____ dollars".
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Rewrite the expression as a product of two factors x²+x-2
we will multiply x by both terms in the parentheses, giving us x²+2x. Then, we will subtract 1 from this expression, leaving us with x²+2x-1. Therefore, the expanded expression is x²+2x-1.
x(x+2) - 1To rewrite the expression x²+x-2 as a product of two factors, we will use algebraic factoring. First, we will factor out the greatest common factor from the expression, which is x. We do this by dividing each term in the expression by x, leaving us with x+2 and -1. The expression can now be written as x(x+2) - 1. This expression is the product of two factors x and (x+2) - 1.
x²+2x-1 To expand the expression x(x+2) - 1, we will use the distributive property. First, we will multiply x by both terms in the parentheses, giving us x²+2x. Then, we will subtract 1 from this expression, leaving us with x²+2x-1. Therefore, the expanded expression is x²+2x-1.
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Solve for a.
18= a/2
In the figure below, quadrilateral UVWX is a parallelogram.
Part a) What are the values of p, UV, and VW?
Part b) What property of a parallelogram did you use to solve?
For the given quadrilateral the value of p is 8, UV = 70 and VW = 34. The property of parallelogram used is the opposite sides are parallel and equal.
What is parallelogram and its properties?A quadrilateral having two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Also, the interior angles that are additional to the transversal on the same side. 360 degrees is the sum of all interior angles.
The opposing sides are equal and parallel. Angles on either side are equal. The subsequent or neighbouring angles are additional. All of the angles will be at right angles if any one of them is a right angle.
We know that the opposite sides of a parallelogram are parallel and equal thus,
8p + 6 = 9p – 2
6 + 2 = 9p – 8p
8 = p
Now, the value of side is:
UV = 8(8) + 6 = 70
The value of side VW is:
VW = 5(8) – 6 = 34
Hence, for the given quadrilateral the value of p is 8, UV = 70 and VW = 34. The property of parallelogram used is the opposite sides are parallel and equal.
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Find the value of $x$ . Write your answer in simplest form.
A right triangle is shown. The measure of one of the interior angles is 45 degrees. The length of one of the legs is 1 unit. The length of the hypotenuse is x units.
$x=$
Answer:
The length of the hypotenuse is $x=\sqrt{2}$ units. ($x=\√2$)
Explanation:
If one of the angles of a right triangle is 45 degrees, then the other acute angle is also 45 degrees. Let's label the other leg of the triangle as $y$ units.
Using the Pythagorean Theorem, we have:
$1^2 + y^2 = x^2$
Simplifying:
$1 + y^2 = x^2$
We can also use the fact that the ratio of the sides of a 45-45-90 triangle is $1:1:\sqrt{2}$ to write:
$\frac{x}{1}=\sqrt{2}$
Simplifying:
$x=\sqrt{2}$
Now, substituting this value of $x$ into our equation $1 + y^2 = x^2$, we have:
$1 + y^2 = (\sqrt{2})^2$
Simplifying:
$1 + y^2 = 2$
$y^2 = 1$
$y = 1$ (since we're looking for a positive length)
Therefore, the length of the hypotenuse is $x=\sqrt{2}$ units.
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Deandre wants to measure the height of a tree. He sights the top of the tree, using a mirror that is lying flat on the ground. The mirror is 33 ft from the tree, and Deandre is standing 10.2 ft from the mirror, as shown in the figure. His eyes are 6 ft above the ground. How tall is the tree? Round your answer to the nearest foot.
Question Darryl plays a card game with his sister. He uses a 52 card deck and she chooses 1 card. If she chooses the 4 of hearts, she will win $10 and if she chooses any spade, she loses $2. If any other card is drawn, they break even. What is the expected value for Darryl's sister? Round to the nearest cent. Do not round until your final calculation. Note: within a deck of cards, there are 4 equal suits: hearts, clubs, diamonds, and spades.
The expected value for Darryl's sister is -$0.31. This means that on average, she can expect to lose 31 cents per game played with the 52 card deck.
The expected value for Darryl's sister can be calculated by multiplying the probability of each outcome by the value of that outcome and then summing those products.
Probability of choosing the 4 of hearts: 1/52
Probability of choosing any spade: 13/52
Probability of choosing any other card: 38/52
Expected value = (1/52)($10) + (13/52)(-$2) + (38/52)($0)
Expected value = $0.19 - $0.50 + $0
Expected value = -$0.31
Therefore, the expected value for Darryl's sister is -$0.31. This means that on average, she can expect to lose 31 cents per game played with the 52 card deck.
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A chemical company makes two brands of antifreeze. The first brand is 45% pure antifreeze, and the second brand is 70% pure antifreeze. In ord gallons of a mixture that contains 65% pure antifreeze, how many gallons of each brand of antifreeze must be used?
Answer:
5 gallons of 70% antifreeze, 1 gallon of 45% antifreeze
Step-by-step explanation:
Many different combinations can be made to equal 65% pure antifreeze.
Essentially, this problem is asking us to find a combination that fits a mean of 65%. So, multiply 5 by 70 to get 350, and 1 x 45 to get 45. Add 350 + 45 together to get 395, and divide by the total amount of gallons used - 6.
395/6 = 65, which is the target value.
suppose we draw 2500 samples of size 100 from a population and compute a 97% confidence interval for each sample. approximately how many of those intervals will contain the population mean?
There are approximately 2425 of those intervals that will contain the population means.
What is a confidence interval?A confidence interval is a range of values that includes a parameter estimate with a certain degree of certainty. A confidence interval is a measure of uncertainty about an estimation procedure, rather than a statement about a parameter. A confidence interval is a specific value or a range of values in statistics that is used to estimate population characteristics.
This is because a 97% confidence interval means that if we were to repeat the sampling process many times, about 97% of the resulting intervals would contain the population mean. In other words, we would expect 97% of the intervals to be "correct" in the long run, while about 3% of them would not contain the population mean.
Therefore, out of the 2500 intervals we compute, we would expect approximately 0.97 x 2500 = 2425 intervals to contain the population mean, and about 0.03 x 2500 = 75 intervals not to contain the population mean. However, it's important to note that the actual number of intervals that contain the population means may vary from this expected value due to random sampling variability.
Where, [tex]\[\bar{x}\][/tex] is the sample mean
[tex]\[\frac{s}{\sqrt{n}}\][/tex] is the standard error of the mean
zα/2 is the z-score that covers α/2 in the tails of a normal distribution.
As we draw 2500 samples of size 100 from a population and compute a 97% confidence interval for each sample, the proportion of the sample will contain the population means.
Therefore, the number of intervals that will contain the population mean is approximately 2425.
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(-7+3)4 with work shown
Answer: -16
Step-by-step explanation:
First, we need to solve the parentheses by performing the operation inside them. (-7 + 3) equals -4, so we can substitute that in the expression to get:
(-7 + 3)4 = -4 × 4
Next, we can perform the multiplication operation:
-4 × 4 = -16
Therefore, (-7 + 3)4 = -16.
Answer:
-16
Step-by-step explanation:
First, you distribute 4 into the equation, basically multiplying, so 4 * 3 is 12, then 4 * -7 is -28. Add them together so -28 + 12 is -16
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Mr. Kelly buys a total of 40 boxes of pens and pencils for his class. Each box of
pens costs $5. Each box of pencils costs $2. Mr. Kelly spends a total of $131 on
the pens and pencils.
Which equations form a system of equations that can be used to determine the
number of boxes of pens, x, and the number of boxes of pencils, y, that Mr. Kelly
buys? Select two correct answers.
A. x+y = 40
B. x+y = 131
C. 5x+2y = 40
D. 2x + 5y = 40
E. 5x + 2y = 131
F.2x + 5y = 131
The correct equations that form a system of equations to determine the number of boxes of pens, x, and the number of boxes of pencils, y, that Mr. Kelly buys are:
A. x+y = 40 (since he buys a total of 40 boxes)
E. 5x + 2y = 131 (since he spends a total of $131)
Therefore, the correct answers are A and E.
13. (6 pts) Jasmine has a cube with an edge length of 4 inches. How would the volume of the cube change if the edge length were doubled?
A) The volume is multiplied by 2
B) The volume is multiplied by 6.
C) The volume is multiplied by 4.
D) The volume is multiplied by 8.
Answer: The volume of a cube is calculated by multiplying the edge length by itself three times, or V = l^3.
If the edge length of Jasmine's cube is doubled, the new edge length will be 4 inches x 2 = 8 inches.
Thus, the new volume of the cube would be V = 8^3 = 512 cubic inches.
Comparing this new volume to the original volume of the cube, V = 4^3 = 64 cubic inches, we can see that the new volume is multiplied by 8.
Therefore, the correct answer is D) The volume is multiplied by 8.
Step-by-step explanation:
1)Sketch the function f(x)=1 /(x−2)^3 −8.
Calculate and show the x-intercepts and y-intercepts, as well as
the horizontal and vertical asymptotes. Must show all the steps in
the calculations in order to receive credit.
2) Calculate the average rate of change of f(x)=2x^3−3x^2−4x
from x=1 to x=3. Must show all the steps in the calculations in order to receive credit.
1) f(x) = 1/(x - 2)^3 - 8
2)f(x) from x = 1 to x = 3 is 15.5.
1) Sketch the function f(x) = 1/(x - 2)^3 - 8 and calculate its x-intercepts, y-intercepts, horizontal and vertical asymptotes.To sketch the function f(x) = 1/(x - 2)^3 - 8, we need to find the x-intercepts, y-intercepts, horizontal and vertical asymptotes. First, let's find the x-intercept. It's a point on the graph where the function intersects the x-axis, so y-coordinate should be zero.f(x) = 1/(x - 2)^3 - 8y = 0 = 1/(x - 2)^3 - 8⇒ 1/(x - 2)^3 = 8⇒ (x - 2)^3 = 1/8⇒ x - 2 = 1/2⇒ x = 2 + 1/2 = 5/2So, the x-intercept is (5/2, 0).Next, let's find the y-intercept. It's a point on the graph where the function intersects the y-axis, so x-coordinate should be zero.f(x) = 1/(x - 2)^3 - 8x = 0 = 1/(0 - 2)^3 - 8⇒ 1/(-2)^3 - 8 = -1/8 - 8 = -8 1/8So, the y-intercept is (0, -8 1/8).Next, let's find the horizontal asymptote. As x approaches infinity or negative infinity, the function approaches a certain value, which is the horizontal asymptote.To find the horizontal asymptote, we need to find the limit of f(x) as x approaches infinity.f(x) = 1/(x - 2)^3 - 8∴ as x → ∞, (x - 2)^3 → ∞⇒ 1/(x - 2)^3 → 0So, the horizontal asymptote is y = -8. To find the vertical asymptotes, we need to find the values of x that make the denominator of the fraction zero.f(x) = 1/(x - 2)^3 - 8x - 2 = 0 = (x - 2)^3⇒ x = 2So, the vertical asymptote is x = 2.Now, we can sketch the function as shown below. f(x) = 1/(x - 2)^3 - 8Answer:2) Calculate the average rate of change of f(x) = 2x^3 - 3x^2 - 4x from x = 1 to x = 3.To find the average rate of change of a function, we need to divide the change in the function by the change in the variable. In this case, the change in the variable is Δx = 3 - 1 = 2.f(x) = 2x^3 - 3x^2 - 4xf(1) = 2(1)^3 - 3(1)^2 - 4(1) = -5f(3) = 2(3)^3 - 3(3)^2 - 4(3) = 26So, the change in the function is Δf = f(3) - f(1) = 26 - (-5) = 31.Now, the average rate of change is given by,Δf/Δx = 31/2 = 15.5Therefore, the average rate of change of f(x) from x = 1 to x = 3 is 15.5.
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