The volume of the solid obtained by rotating the region in the first quadrant bounded by [tex]$y=x^5$[/tex], y = 1, and the y-axis about the line y = −2 is 29.0359321.
The region is bounded in the xy plane and is said to be bounded in between the curve, line and y-axis as given by
Curve: [tex]$y=x^5$[/tex]
Line: y = 1
y-axis: x = 0
The curve and line intersect at [tex]x^5=1[/tex] and x = 1
The region bounded can be expressed as
[tex]x^5 \leq y \leq 1 \\[/tex]
0 ≤ x ≤ 1
Here we have to determine the volume of the solid obtained by rotating the region in the first quadrant bounded by y=x5, y=1
Now if the solid is obtained by the rotation of the region about the line y = -2 then the volume of the solid is evaluated by the method of the rings/washers as
Volume [tex]$=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{A}(\mathrm{x}) \mathrm{dx}[/tex]
Volume [tex]$=\int_{\mathrm{a}}^{\mathrm{b}} \pi\left[\text { (outer radius) }^2-\text { (inner radius }^2\right] \mathrm{dx}[/tex]
Volume [tex]$=\pi \int_0^1\left[(1-(-5))^2-\left(x^5-(-5)\right)^2\right] d x[/tex]
Volume [tex]$=\pi \int_0^1\left[(1+5)^2-\left(x^5+5\right)^2\right] d x[/tex]
Volume [tex]$=\pi \int_0^1\left[(6)^2-\left(x^5+5\right)^2\right] \mathrm{d} x[/tex]
Volume = [tex]$\pi \int_0^1\left[(36)-\left(x^{10}+10 x^5+25\right)\right] d x$$[/tex]
Volume = [tex]$\pi \int_0^1\left(36-x^{10}-10 x^5-25\right) d x[/tex]
Volume [tex]$=\pi \int_0^1\left(11-x^{10}-10 x^5\right) d x[/tex]
Volume [tex]$=\pi\left[11 x-\frac{1}{11} x^{11}-10\left(\frac{x^6}{6}\right)\right]_0^1$$$$[/tex]
Volume [tex]$=\pi\left[11 x-\frac{1}{11} x^{11}-\frac{5}{3} x^6\right]_0^1$$$$[/tex]
Volume [tex]$=\pi\left(11(1-0)-\frac{1}{11}\left(1^{11}-0\right)-\frac{5}{3}\left(1^6-0\right)\right)[/tex]
Volume [tex]=\pi\left(11(1)-\frac{1}{11}(1)-\frac{5}{3}(1)\right)[/tex]
Volume [tex]=\pi\left(11-\frac{1}{11}-\frac{5}{3}\right)$$[/tex]
Volume = [tex]$\frac{305}{33} \pi[/tex]
Therefore the volume of the solid is
V [tex]=\frac{305}{33} \pi \approx[/tex] 29.0359321
Therefore the volume of the solid is 29.0359321
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To conserve water, many communities have developed water restrictions. The water utility charges a fee of $29, plus an additional $1.41 per hundred cubic feet (HCF) of water. The recommended monthly bill for a household is between $54 and $82 dollars per month. If x represents the water usage in HCF in a household, write a compound inequality to represent the scenario and then determine the recommended range of water consumption. (Round your answer to one decimal place.)
Answer:
I can lead you with this equation: 1.41x + 29 = Cost
I think it will be easy from here on for you.
Step-by-step explanation:
Clare says, “This classroom is 11 meters long. A meter is longer than a yard, so if I measure the length of this classroom in yards, I will get less than 11 yards.” Do you agree with Clare? Explain your reasoning.
Please help me with this question
The measure of angle 2 from the diagram is equivalent to 111 degrees
How to solve for an unknown angle in a line geometry
An angle is defined as the point where two lines meet. From the diagram shown the measure of <1 and <2 lies on the same straight line.
Since the sum of angles on a straight line is 180 degrees, hence;
<1 + < 2 = 180
Substitute the given angle
69 + <2 = 180
<2 = 180 - 69
<2 = 111 degrees
Hence the measure of <2 from the given diagram is 111 degrees.
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suppose a third row of tiles identical to the ones above is added to the model. How does that change the two expressions? Please helpp!
Two equivalent expressions for the model include the following:
4 + 6x.
2(2 + 3x).
Assuming a third row of tiles identical to the ones above is added to the model, the two expressions becomes;
6 + 9x.
3(2 + 3x).
What is an expression?In Mathematics, an expression is sometimes referred to as an equation and it can be defined as a mathematical equation which is typically used for illustrating the relationship that exist between two (2) or more variables and numerical quantities (number).
Based on the information provided about the algebra tile model, we can logically deduce the following:
Positive signs (+) = 4
Negative signs (-) = 6
Expression = 4 + 6x
By factoring, we have:
4 + 6x = 2(2 + 3x).
By adding a third row of tiles identical to the ones above, we have the following expression:
Positive signs (+) = 6
Negative signs (-) = 9
Expression = 4 + 6x + 2 + 3x = 6 + 9x
By factoring, we have:
6 + 9x = 3(2 + 3x).
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10. Order the following from least to greatest and then place them in the correct locat
{-3 1/3, 9/2,√15, 16/5 , √6
Answer:
Step-by-step explanation:
-3 1/3 goes to between -3 and -4
9/2=4 1/2 between 4 and 5
[tex]\sqrt{15}[/tex] between 3 and 4
16/5=3 1/5 between 3 and 4
\sqrt{15} between 2 and 3
Poor fitness in adolescents and adults increases the risk of cardiovascular disease. In a study of 3110 adolescents and 2205 adults, researches found 33.9% of adolescents and 14% of adults were unfit; the percentage was similar in adolescent males 32.5%) and females 35%), but was higher in adult females 16.2%) than in adult then in adult males 11.2%) Chart of Percentage Unfit vs Age Group, Gender 35 30 25 20 15 10 Gender Age Group A B C A B C Adult (a) Match the bars on the chart with the appropriate category from the table Bar B Overall Bar C Male v Bar A Femalev (b) Comment on the interesting features of your graphical display
(a) Bar A: A, Bar B: O, Bar C: F
(b) One interesting feature of the graphical display is the clear difference in the percentage of individuals who are considered unfit between adolescents and adults.
(a) Bar A: Adolescents
Bar B: Overall (both adolescents and adults)
Bar C: Adult Males vs. Adult Females
(b) The percentage of unfit individuals is much higher in adolescents (33.9%) than in adults (14%). Additionally, within the adult group, the percentage of unfit individuals is higher in females (16.2%) than in males (11.2%). Another interesting feature is that there is no significant difference in the percentage of unfit individuals between male and female adolescents.
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Use the graph below to determine the number of solutions the system has. X=4. Y=x+3
The system of linear equations x = 4 and y = -x -1 has a unique solution.
What is linear equation?A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation.
Given is a graph of lines,
We need to find the number of solutions do the lines x = 4 and y = -x-1 have,
In the graph of the lines x = 4 and y = -x -1, intersecting at a point, we know that the lines which intersects at a single point will have only one solution.
Also,
The system of equation has the solution at the point where the line intersects,
By observing the graph it can be concluded that the graph of x = 4 and y = - x - 1 intersect only at one point i.e (4, -5).
Therefore, the solution of the given system of equations is (4, -5)
Hence, the system of linear equations x = 4 and y = -x -1 has a unique solution.
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Find the product.
112⋅213
Write the answer as a mixed number.
Answer:3 1/2
[tex]\frac{3}{2} times\frac{7}{3} =\frac{21}{6} =3\frac{3}{6} = 3\frac{1}{2}\\[/tex]
Step-by-step explanation:Someone help me with these math problems
Answer: this doesnt seem complete, like what are we supposed to explain with that
Step-by-step explanation:
. A triangle has vertices on a coordinate grid at points J(-5,6), K(5, 6), and L(-5, -5). What is the length, in units, of JK?
According to the given information the answer is JK thus has a 10-unit length.
How length is an example?The measurement or size of a thing end to end is referred to as its length. In other words, it is the larger of an object's elevated two or three geometric dimensions. For instance, a rectangle's dimensions are determined by its length and width.
We must determine the separation among points J and K in order to get the length of JK. The distance formula is as follows:
d = √((x2 - x1)² + (y2 - y1)²)
Where (x1, y1) = (-5, 6) is the coordinate of J and (x2, y2) = (5, 6) is the coordinate of K.
Plugging in the values, we get:
d = √(5 - (-5))² + (6 - 6)²)
= √(10² + 0²)
= √100
= 10
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Applications of Quadratics. Only 2 questions please help.
Answer: look at the images for the answer ( the first image is for #1 and the second image is for #2 )
Step-by-step explanation:
ben bought six and three fifths bags of rocks to use in his garden. He used four and one fifth bags. How many bags does Ben have left?
Answer:
To find the answer, we need to subtract the amount of bags Ben used from the amount he bought:
6 and 3/5 - 4 and 1/5
To subtract fractions, we need to have a common denominator. The smallest common denominator for 5 and 20 is 20, so we'll convert both fractions:
6 and 3/5 = 6 * 5/5 + 3/5 = 30/5 + 3/5 = 33/5
4 and 1/5 = 4 * 20/20 + 1/5 = 80/20 + 1/5 = 81/20
Now we can subtract:
33/5 - 81/20
We need to convert the mixed number to an improper fraction and find a common denominator:
33/5 - 81/20 = 132/20 - 81/20
Now we can subtract:
132/20 - 81/20 = 51/20
So Ben has 51/20 bags of rocks left. This can be simplified to 2 and 11/20 bags.
Answer:
2 2/5
Step-by-step explanation:
6 3/5 - 4 1/5 = 2 2/5
If the cost, C(x), for manufacturing x units of a certain product is given by
C(x) = x² - 9x + 30
find the number of units manufactured at a cost of $8200.
The number of units manufactured at the cost of $8200 will be 95.
A quadratic equation is a polynomial with a degree of 2 or the maximum power of the variable is 2 in quadratic equations. It has two solutions as its maximum power is 2.
Given that the cost, C(x), for manufacturing x units of a certain product is
C(x) = x² - 9x + 30.
The number of units will be calculated as:-
C(x) = x² - 9x + 30
8200 = x² - 9x + 30
x² - 9x - 8170 = 0
Solve the equation as below,
x² - 9x - 8170 = 0
x² - 95x + 86x - 8170 = 0
x ( x - 95 ) + 86 ( x - 95 ) = 0
( x - 95 ) ( x + 86 ) = 0
x = 95 units
Therefore, the number of units manufactured at the cost of $8200 will be 95.
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A man buys a house for $200, 000. He makes a $50, 000 down payment over the next 10 years. If the interest on the debt 12%, compounded quarterly find (1)size of the quarterly payment. (2) Total amount of the payment and (3) Total amount of interest paid.
To solve this problem, we can use the formula for calculating the quarterly payment of a loan with compound interest:
Payment = (r * P) / (1 - (1 + r)^(-n))
where:
r = quarterly interest rate (12% / 4 = 0.03)
P = principal amount (remaining debt after down payment) = $150,000
n = total number of quarterly payments (10 years * 4 quarters per year = 40)
(1) Calculation of Quarterly Payment:
Payment = (0.03 * 150,000) / (1 - (1 + 0.03)^(-40)) = $2,365.87 (rounded to the nearest cent)
Therefore, the quarterly payment is $2,365.87.
(2) Calculation of Total Amount of Payment:
Total amount of payment = (number of payments) * (quarterly payment)
number of payments = 10 years * 4 quarters per year = 40
Total amount of payment = 40 * 2,365.87 = $94,634.80
Therefore, the total amount of payment is $94,634.80.
(3) Calculation of Total Amount of Interest Paid:
Total amount of interest paid = total amount of payment - principal amount
Total amount of interest paid = $94,634.80 - $150,000 = -$55,365.20
Note that the result is negative because the down payment ($50,000) exceeds the total amount of interest paid over the life of the loan. Therefore, in this case, the total amount of interest paid is $0, and the man ends up paying a total of $100,000 ($50,000 down payment + $50,000 in quarterly payments).
The length of the sides of two squares differ by 3cm and the sum of their areas is 317cm^2 find the length of the sides of the two squares.
The side lengths of the given squares are 11 cm and 14 cm
What is a square?A square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides.
Given that, the sides of two squares differ by 3cm and the sum of their areas is 317cm², we need to find the length of the sides of the two squares.
Let the sides of the squares be x and y, we know that the area of a square is their side's square.
Therefore, establishing the system of equations,
x-y = 3
x = y+3.......(i)
x²+y² = 317....(ii)
Put eq(i) in eq(ii)
(y+3)²+y² = 317
y²+9+6y+y² = 317
2y²+6y-308 = 0
On factorizing, we get,
y = 11 and y = -14
Since, the length can not be negative so, ignore -14
Put y = 11 in eq(i)
x = 11+3
x = 14
Therefore, the side lengths of the given squares are 11 cm and 14 cm
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use the combined gas law equation to determine the final volume of a system initially of 2 liters if the pressure is tripled and the temperature is tripled.
The final volume of a system with initially of 2 liters if the pressure is tripled and the temperature is tripled, is 11.57 liters.
The combined gas law equation relates the initial and final conditions of a system of gas by considering the effect of changes in pressure, volume, and temperature:
P1 * V1 / T1 = P2 * V2 / T2
Given the initial conditions of the system:
V1 = 2 liters
P1 = 1 atm
T1 = 273 K (room temperature)
And the changes in pressure and temperature:
P2 = 3 * P1 = 3 atm
T2 = 3 * T1 = 819 K
We can calculate the final volume of the system:
V2 = (P1 * V1 / T1) * (T2 / P2) = (1 atm * 2 liters / 273 K) * (819 K / 3 atm) = approximately 11.57 liters
So the final volume of the system would be approximately 11.57 liters if the pressure is tripled and the temperature is tripled from the initial conditions.
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consider the following proof attempt for conjecture : 1. - premise 2. - existential instantiation for (1) 3. - simplification rule for (2) 4. - universal generalization for (3) mark all statements that describe errors in this proof.
Conjectures are unsubstantiated claims. A theorem is created when someone proves a conjecture. In logic types that deal with conjunctions, the rule of simplification is a valid argument. The rule of inference that states that ∀xP(x) is true given the premise that P(c) is true for all elements c in the domain is known as universal generalization.
The Conjecture Rule
Conjectures are unsubstantiated claims. A theorem is created when someone proves a conjecture.
We seek to understand a conjecture on three levels: we want to know what it means, we want to know why we think the claim is true, and we want to know how it fits into a larger set of ideas.
Rule for Simplification
In logic types that deal with conjunctions, the rule of simplification is a valid argument.
This includes propositional and predicate logic, as well as natural deduction.
(1) Proof Rule: If we can reach a conclusion ϕ∧ψ, we can infer ϕ.
(2): If we can reach a conclusion ϕ∧ψ, we can infer ψ.
Rule of Universal Generalization
The rule of inference that states that ∀xP(x) is true given the premise that P(c) is true for all elements c in the domain is known as universal generalization. When we show that ∀xP(x) is true by taking an arbitrary element c from the domain and showing that P(c) is true, we are using universal generalization. The element c that we choose must be arbitrary and not a specific domain element.
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The complete question is:
Write about the conjecture rule, simplification rule, universal generalization rule.
Can someone please help me for 50 points and brainliest and also show your work on paper and upload the picture
wrong answer such as guessing, or even putting random answers will be reported !
answer only even numbers
What is the average rate of change of f(x) over the x-interval [8,64] for the function
f(x) = 10\sqrt[3]{x}-5 (if equation doesn't make sense, copy/paste it into a math solver. I have the answer, i just need an in-depth answer to how to get the answer pleaseee)
The average rate of change of the function, [tex]f(x) = 10\sqrt[3]{x}-5[/tex] over the interval of x=8 to x=64 is calculated as: 0.357.
How to Find the Average Rate of Change of a Function?The average rate at which one quantity changes in relation to another's change is referred to as the average rate of change function.
A method that determines the amount of change in one item divided by the corresponding amount of change in another is known as an average rate of change function.
The average rate of change = f(b) - f(a) / b - a.
Given the following:
The function is: [tex]f(x) = 10\sqrt[3]{x}-5[/tex]
Interval is: x = 8 to x = 64
Therefore:
a = 8
b = 64
f(a) = f(8) =
[tex]f(x) = 10\sqrt[3]{8}-5\\\\f(x)=10*(2)-5\\\\f(x)=20-5\\\\f(x)=15[/tex]
f(a) = f(8) = 15
f(b) = f(10)
[tex]f(x) = 10\sqrt[3]{64}-5\\\\f(x)=10*(4)-5\\\\f(x)=40-5\\\\f(x)=35[/tex]
f(b) = f(64) = 35
Therefore:
Average rate of change =
[tex]=\frac{f(b)-f(a)}{b-a}\\\\=\frac{35-15}{64-8} \\\\=\frac{20}{56} \\\\=0.357[/tex]
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Cecily purchases a box of 100 paper clips. She puts 37 /100 of the paper clips in a jar on her desk and puts another 6 /10 in her drawer at home. Shade a grid that shows how many of the paper clips are in Cecily's jar and drawer, then write the fraction tbe grid represents.
If Cecily purchases a box of 100 paper clips. She puts 37 /100 of the paper clips in a jar on her desk and puts another 6 /10 in her drawer at home. The number of the paper clips that are in Cecily's jar is: 43.
How to find the number of the paper clips?Here is a grid to show the number of paper clips that Cecily has in her jar and drawer:
JAR | DRAWER
|
37 | 60
|
100 | 100
The fraction that the grid represents is :
37/100 + 6/100 = 43/100.
Therefore, Cecily has 43 out of 100 paper clips in her jar and drawer.
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A box contains two coins: a regular coin and one two-headed coin. I choose a coin at random and toss it twice. Define the following events.
A= First toss results in a H
B= Second toss results in a H
C= Regular coin has been selected
Are A and B independent? Are A and B conditionally independent given C? Find P(A|C), P(B|C), P(A∩B|C), P(A), P(B), and P(A∩B), and use them to answer the question.
A and B are not independent since the result of one toss affects the second toss. Specifically, if the first toss result is a head (A), then the second toss will be more likely to result in a head (B).
A and B are conditionally independent given C since the probability of getting head(A) for the first toss does not depend on the type of coin selected (C).
P(A|C) = 1/2, P(B|C) = 1/2, P(A∩B|C) = 1/4, P(A) = 1/2, P(B) = 3/4, and P(A∩B) = 1/4. Therefore, given the information that a regular coin has been selected (C), the probability of A and B occurring independently is 1/4. This suggests that A and B are conditionally independent given C.
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Given that h(x)=x2, find (h•h)(x)
Compare the total number of students that attended 6 or more baseball games to the number of students that attended 1 or less
The ratio will be equal to 13:15 for the students who attended the games.
What is a ratio?The ratio actually compares the data of two numbers. It identifies that the one data is how many times the other data.
Melanie randomly selected 15 seventh-grade students and asked them how many soccer games they had attended this school year. Each student's number of games attended is listed below 3. 2 , 6, 4, 7, 0, 4, 5, 0, 1, 2, 3, 2, 1, 4
Here, the number of students who did not attendgames=2 [since there are 2 0s]
The total number of students = 15
Number of students attending games= 15-2=13
The ratio will be equal to 13: 15
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The complete question is given below.
Melanie asked 15 random seventh-grade students how many soccer games they attended this school year. The number of games attended by each student is listed below. 2, 3, 6, 4, 7, 0, 4, 5, 0, 1, 2, 3, 2, 1, 4 What is the ratio of the total number of surveyed students who attended soccer games to the total number of students surveyed?
Earline needs to save $367.50 for her summer vacation. She plans on saving $52.50 per week. In 6 weeks, will she have enough money? Explain.
Hint: write a multiplication equation
solve the following system of equations graphics on the set of axes below
Answer:
(1, -6)
Step-by-step explanation:
3x + y = -3
y = -3x - 3
Let p= "x>7,"q="x=7," and r="12>x." Select the symbolic form for each of the following statements. (a) x≥7p∨rp∼qp∨qp∧qq∼r (b) 12>x>7r∧pr∼pr∨pp∧qp∼q(c) 12>x≥7p∧(q∨r)r∧(p∨q)r∧(p∼q)r∨(p∧q)r∼(p∨q)
(a) p ∨ r: "x≥7 or 12>x.", (b) r ∧ p: "12>x and x>7." and (c) r ∧ (q ∨ p) : "12>x and (x=7 or x>7),
" which simplifies to "12>x." The second part, (p ∨ q) ∧ r ∨ (p ∼ q) ∧ r ∨ (p ∧ q) ∧ r ∼ (p ∨ q), is always true when r is true, so it is equivalent to r. Thus, the entire statement is equivalent to "12>x and (x=7 or x>7)."
(a) p ∨ r: This statement is a logical disjunction, also known as an "or" statement. It is true if either the left side (p) is true or the right side (r) is true, or both. The statement "x>7" is equivalent to "x≥7 or x=7", so the statement p ∨ r can be read as "x≥7 or 12>x."
(b) r ∧ p: This statement is a logical conjunction, also known as an "and" statement. It is true only if both the left side (r) and the right side (p) are true. The statement "12>x" is equivalent to "x<12", so the statement r can be read as "x<12." Combining this with the statement p (x>7) gives "12>x>7."
(c) r ∧ (q ∨ p): This statement is a conjunction of r and a logical disjunction (q ∨ p). As we saw in part (a), p is equivalent to "x≥7 or x=7", so the statement (q ∨ p) can be read as "x=7 or x≥7". Combining this with the statement r (12>x) gives "12>x and (x=7 or x>7)", which can be simplified to "12>x."
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The program below is intended to count the number of prime numbers in a list called numbers and display the result. The program uses the procedure isPrime (n), which returns true if n is a prime number
and false otherwise.
The program does not work as intended.
Which two lines of code should be removed so that the program will work as intended? Select two answers.
The first line that should be removed is the print statement. The code should not display the result right away, as this will prevent it from counting the number of prime numbers in the list.
The second line of code that should be removed is the second statement, which checks to see if the index is equal to 0. This line of code is unnecessary, as the prime numbers are already tested in the if statement before it.
In order for the program to work as intended, the code should only include the if statement that checks to see if the number is a prime number, the for loop that iterates through the list and increments the counter for each prime number found, and the print statement that displays the result at the end. This will ensure that the program counts the number of prime numbers in the list and displays the result correctly.
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A headboard for a bed is shaped like an isosceles trapezoid. The length of one leg of
the trapezoid is 6x + 18 inches and the length of the other leg is 8x + 6 inches.
What is the length of the legs?
Length of the sides is 54 and 54 inches.
What is length?Length is defined as the measurement of distance of an object from one end to the other.
A headboard for a bed is shaped like an isosceles trapezoid.
Since it is a isosceles trapezoid the length are equal in number.
The we have to equate the given lengths,
i.e., 6x+18 = 8x+6
Rearranging the terms we get,
6x-8x = 6-18
-2x = -12
Divide by -2 on both sides we get,
x = 6
Sub x in the given lengths we get,
6x + 18 ⇒ 6 * 6 +18 = 54
8x + 6 ⇒8 *6 +6 = 54
So, length of the sides is 54 inches.
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what is a divided by 6 -11=25
You can solve this by using the addition and multiplication properties of equality
the original equation is:
a divided by 6 - 11 = 25
then you can add 11 to both sides to get rid of it (addition property of equality)
a divided by 6 = 36
then you use the multiplication property of equality
a = 216
the amount that households pay service providers for access to the internet varies quite a bit, but the mean monthly fee is $28 and the standard deviation is $10. the distribution is not normal: many households pay about $10 for limited dial-up access or about $30 for unlimited dial-up access, but some pay much more for faster connections. a sample survey asks an srs of 500 households with internet access how much they pay. let x be the mean amount paid.a. Explain why you can't determine the probability that the amount a randomly selected household pays for access to the Internet exceeds 55 .$b. What are the mean and standard deviation of the sampling distribution of x^-x ?c. What is the shape of the sampling distribution of x^-x ? Justify your answer.d. Find the probability that the average fee paid by the sample of households exceeds 55 .$
a) We cannot determine the probability
b) The mean of x - 55 is $28 - $55 = -$27, and the standard deviation of x - 55 is $0.45.
c) By the central limit theorem, the sampling distribution of x^-x will be approximately normal if the sample size is large enough.
d) The probability that the average fee paid by the sample of households exceeds $55 is very close to 0.
a. We cannot determine the probability that the amount a randomly selected household pays for access to the internet exceeds $55 because we don't have information about the distribution of individual payments, only the mean and standard deviation of the population.
b. The mean of the sampling distribution of x is equal to the population mean, which is $28. The standard deviation of the sampling distribution of x is equal to the population standard deviation divided by the square root of the sample size, which is $10/sqrt(500) ≈ $0.45. Therefore, the mean of x - 55 is $28 - $55 = -$27, and the standard deviation of x - 55 is $0.45.
c. By the central limit theorem, the sampling distribution of x^-x will be approximately normal if the sample size is large enough. In this case, since the sample size is 500 and greater than 30, we can assume that the sampling distribution is approximately normal.
d. To find the probability that the average fee paid by the sample of households exceeds $55, we need to standardize the value of x using the sampling distribution of x^-x and then find the probability of a z-score greater than or equal to:
z = (55 - 28)/0.45 ≈ 60. The probability of a z-score greater than or equal to 60 is extremely small, as the normal distribution tails off quickly. Therefore, the probability that the average fee paid by the sample of households exceeds $55 is very close to 0.
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