Answer:
8 in³
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{4 cm}\underline{Volume of a sphere}\\\\$V=\dfrac{4}{3} \pi r^3$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \end{minipage}}[/tex]
Since "hemi" is a prefix meaning one half, the volume of a hemisphere is one half the volume of a sphere:
[tex]\implies V_{\sf hemisphere}=\dfrac{\frac{4}{3}\pi r^3}{2}[/tex]
[tex]\implies V_{\sf hemisphere}=\dfrac{2}{3}\pi r^3[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Volume of a cone}\\\\$V=\dfrac{1}{3} \pi r^2 h$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]
Given the diameter of the ice cream cone is equal to its height:
diameter = h = 2rSubstitute h = 2r into the equation for the volume of the cone:
[tex]\implies V_{\sf cone}=\dfrac{1}{3} \pi r^2 \cdot 2r[/tex]
[tex]\implies V_{\sf cone}=\dfrac{2}{3} \pi r^3[/tex]
Therefore, we can see that the formula for the volume of ice cream in the cone is the same as the formula for the volume of the hemisphere of ice cream.
This means that the volume of the cone of ice cream is equal to the volume of the hemisphere of ice cream.
Given the volume of the hemisphere is 4 in³, the total volume of the ice cream is 8 in³.
7. Return on equity will not change if the firm increases its use of debt.
The statement "Return on equity will not change if the firm increases its use of debt" is false.
What is return in equity?Return on equity (ROE) is a financial ratio that shows how well a company is managing the capital that shareholders have invested in it. To calculate ROE, one would divide net income by shareholder equity.
Given is a statement, "Return on equity will not change if the firm increases its use of debt".
As we know that, The equity equals the assets minus total debt, a company decreases its equity by increasing the debt, or we can say, when debt increases, the equity decreases and since equity is ROE's denominator, ROE in turns gets a boost.
Hence, the statement "Return on equity will not change if the firm increases its use of debt" is false.
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A line on a graph passes through the points (15,10) and (60,20) what is the slope of the line
Answer:
The slope is 2/9.
Step-by-step explanation:
(15, 10) and (60, 20)
m = 20-10/60-15
m = 10/45
m = 2/9
people throw away 12% of what they buy at the grocery store (reader's digest, march 2009). assume this is the true population proportion and that you plan to take a sample survey of 540 grocery shoppers to further investigate their behavior. given this is an infinite population calculate the standard error of the proportion to 4 decimals.
The standard error of the proportion for the given distribution is 0.0139. This means that the probability that sample proportion will differ from the population proportion is 0.0139.
Here, we are given that people throw away 12% of what they buy at the grocery store
⇒ Probability of occurrence of the event (p) = 0.12
Probability of non occurrence of the event (q) = 1 - 0.12 = 0.88
Further, survey is conducted for 540 shoppers,
⇒ Sample size (n) = 540
Now, we know that the standard error (SE) can be calculated using the given formula-
SE =√(p × q/n)
Substituting the values, we get-
SE =√(0.12 × 0.88/ 540)
SE = 0.0139
Thus, we find that the standard error of the proportion for the given distribution is 0.0139
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19 points all i have really..Gavin is hoping to buy his first car in 3 years. He is considering two options to help him save and grow his money. If he currently has $800, should he deposit his money into a savings account that is compounded annually at an interest rate of 9% or in a savings account that is compounded quarterly at an interest rate of 3%?
Gavin should deposit his money into a savings account that is compounded annually at an interest rate of 9%.
What is compound interest?
The interest earned on savings that is computed using both the original principal and the interest accrued over time is known as compound interest.
Given that the current balance is $800.
There were two types of schemes.
First scheme offer compounded annually at an interest rate of 9%
Second scheme offer compounded quarterly at an interest rate of 9%.
The formula of compound interest annually is
A = P(1+r)^t
P = principal = 800
r = rate of interest = 9% = 0.09
t = time = 3 year
A = 800(1+0.09)³
A = 1036.02.
The formula of compound interest annually is
A = P(1+(r/n))^tn
P = principal = 800
r = rate of interest = 3% = 0.03
t = time = 3 year
n = 4
nt = 4×3 =12
A = 800(1+0.03/4)¹²
A = 875.04
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What integer is tripled when 9 is added to three-fourths of it?
Answer:
4
Step-by-step explanation:
let n be the integer , then n tripled is 3n, so
[tex]\frac{3}{4}[/tex] n + 9 = 3n ( multiply through by 4 to clear the fraction )
3n + 36 = 12n ( subtract 3n from both sides )
36 = 9n ( divide both sides by 9 )
4 = n
the required integer is 4
PLEASE HELP IM GOING TO FAIL
4. What is the graph of the system? (1 point)
y≤x +4
2x+y≤-4
Answer:
A
Step-by-step explanation:
The answer is option A
Hope this helps :)
The data below shows the numbers of hours Mrs. Diaz works at the library each month fir eight months. The mean of the data set is 22 hours. What is the mean absolute deviation of the data
The mean absolute deviation of the data 20, 12, 27, 23, 18, 20, 30, and 26 will be 4.5.
What is the mean absolute deviation?It is the average distance between each data point and the mean.
The mean absolute deviation is given as
[tex]\rm MAD = \dfrac{\Sigma _{i = 1}^n|x_i - \mu|}{n}[/tex]
The information underneath shows the quantities of hours Mrs. Diaz chipped in at the library every month for a very long time.
20, 12, 27, 23, 18, 20, 30, 26
The mean of the informational index is 22 hours.
Then the mean absolute deviation of the data will be given as,
MAD = [|20 - 22| + |12 - 22| + |27 - 22| + ... + |30 - 22| + |26 - 22|] / 8
MAD = (2 + 10 + 5 + 1 + 4 + 2 + 8 + 4) / 8
MAD = 36 / 8
MAD = 4.5
The mean absolute deviation of the data will be 4.5.
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The complete question is given below.
The data below show the numbers of hours Mrs. Diaz volunteered at the library each month
for eight months.
20, 12, 27, 23, 18, 20, 30, 26
The mean of the data set is 22 hours. What is the mean absolute deviation of the data?
I need help with my math homework
The multiplications, applying the associative property, are given as follows:
1. (5 x 2) x 3 = 10 x 3 = 30.
2. 4 x (3 x 3) = 4 x 9 = 36.
3. (6 x 1) x 4 = 6 x 4 = 24.
4. 5 x (2 x 4) = 5 x 8 = 40.
5. (2 x 2) x 8 = 4 x 8 = 32.
6. 3 x (2 x 2) = 3 x 4 = 12.
7. (4 x 2) x 4 = 8 x 4 = 32.
8. 6 x (3 x 3) = 6 x 9 = 54.
What is the associative property of multiplication?The associative property of multiplication defines that the order in which the factors are multuiplied does not change the product.
In this problem, we have to consider that the parenthesis gives the precedence of the factors, meaning that the terms inside the parenthesis are multiplied first, and their result should be placed on the blanks below the parenthesis.
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Examine the triangles below, how do we know the triangles are congruent?
Please help it if for a test worth 105 points
The triangles are congruent by SAS rule, according to the question.
What do you mean by congruency means?
When two geometrical figures are completely same, they are said to be congruent. Congruent refers to objects that are the same size and shape. The concept of congruence can be applied to figures, angles, and line segments. Any two line segments are deemed to be congruent if their lengths are the same. If the measure of two angles is the same, they are said to be congruent. The sides and angles of two triangles are said to be equal if they are congruent. Congruent refers to objects that are precisely similar in size and shape. The forms stay equal no matter how they are rotated, flipped, or turned. For instance, create two circles with the same radius, cut them out, and stack them.
According to the given question,
We have the given information:
AC║EC
BC║DC
Now, we need to show the congruency in the given triangles,
We already have two sides of the triangles are parallel.
The angles are also same, when two lines are bisects each other.
∠ACB≈∠ACD
Therefore, the given triangles are congruent by SAS rule.
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Question 2 of 10 A tessellation could be created using circles like the one shown below. A. True B. False
Answer:
B. False
Step-by-step explanation:
You want to know if it is true that a tessellation can be created from circles.
TessellationA tessellation is the covering of a plane, using one or more geometric shapes with no overlaps and no gaps.
As the attachment shows, it is not possible to cover a plane with circles, leaving no gaps.
A tessellation cannot be created using circles. The proposition is False.
During one flu epidemic the proportion of adults nationwide who have come down with the flu is 8%. The manufacturer of a flu vaccine claims that those who have been vaccinated are less likely to catch the flu.
The hypotheses are as follows:
Null hypothesis: proportion of those vaccinated catching the flu = 8%
Alternative hypothesis: proportion of those vaccinated catching the flu < 8%
Select one:
A.
Rejecting the null hypothesis means there is evidence that the proportion of those vaccinated catching the flu is less than 8%
Failing to reject the null hypothesis means there is insufficient evidence to conclude that the proportion of those vaccinated catching the flu is less than 8%.
B.
Rejecting the null hypothesis means there is evidence that the proportion of those vaccinated catching the flu is not equal to 8%
Failing to reject the null hypothesis means there is insufficient evidence to conclude that the proportion of those vaccinated catching the flu is equal to 8%.
C.
Rejecting the null hypothesis means there is insufficient evidence that the proportion of those vaccinated catching the flu is equal to 8%
Accepting the null hypothesis means there is evidence to conclude that the proportion of those vaccinated catching the flu is equal to 8%.
D.
Rejecting the null hypothesis means there is evidence that the proportion of those vaccinated catching the flu is less than 8%
Accepting the null hypothesis means there is evidence to conclude that the proportion of those vaccinated catching the flu is equal to 8%.
The test hypothesis is
H₀: p=0.08 (null hypothesis) and
Hₐ: p<0.08 (alternative hypothesis )
Given that,
8% of adults in the country contract the flu at any given time during a flu epidemic. The following are the hypotheses:
0% of people who received the vaccine contracted the flu.
8% or less of people who received the vaccine ended up getting the flu.
In order to reject the null hypothesis, there must be proof that 8% of individuals who received the vaccine did not also get the flu.
If the null hypothesis is not rejected, then there is insufficient data to draw the inference that the proportion of people who receive the flu shot is 8%.
The test hypothesis is H₀: p=0.08 (null hypothesis) and Hₐ: p<0.08 (alternative hypothesis )
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Write a proof for the following situation. This proof can be a paragraph, flow chart proof or two column proof.
No matter which one you use, the proof should contain both statements that logically lead to the desired conclusion, as well as, the reasons those statements are justified. A two column proof has been started for you.
Lines p and q are parallel and angles 1 and 9 are congruent because they are corresponding angles. Angles 1 and 11 are congruent by given. Since angles 9 and 11 are congruent to the same angle, angles 1, angles 9 and 11 are congruent by transitive or substitution. Lines a and b are parallel.
Amy is going to spend What she saves from her rent budget on entertainment. How much will she’s have for entertainment each month?
Since Amy's rent is 60 percent of the budget for Rent, Utilities, and Food, she will have $853 to spend on entertainment each month.
How is the rent budget determined?The total budget for rent, utilities, and food for nine months is given as $12,798.
Using the mathematical operation of multiplication, the proportion that belongs to rent alone is $7,678.80, representing 60%.
This rental proportion is divided by 9 to determine the monthly budget for rent, which will then be spent on entertainment.
The total budget for rent, utilities, and food for nine months = $12,798
The percentage of the budget for rent = 60%
The amount for rent for nine months = $7,678.80 ($12,798 x 60%)
The monthly rent amount is $853.20 ($7,678.80/9)
Thus, we can conclude that Amy will spend $853 monthly on entertainment if she saves the rent budget.
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Question Completion:The percentage of the rent to the total rent, utilities, and food budget is 60%.
the quadratic formula will give us the coordinates of the points of intersection of a line and a quadratic only when the value of the discriminant, is .
The quadratic formula will give us the coordinates of the points of intersection of a line and a quadratic only when the value of the discriminant b^2 - 4ac is either positive or zero .
Given :
the quadratic formula will give us the coordinates of the points of intersection of a line and a quadratic only when the value of the discriminant, is .
Quadratic equation :
Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0.
Formula :
= -b ± [tex]\sqrt{b^2 - 4ac / 2a}[/tex]
Discriminant = b^2 - 4ac
if b^2 - 4ac = 0 or positive value then only the formula give us the coordinates of the points of intersection of a line .
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Translate this phrase into an algebraic expression.
19 increased by twice a number
Use the variable n to represent the unknown number.
The algebraic expression represents the result of adding 19 to twice a number is 19 + 2n.
What is an algebraic expression?
An algebraic expression is a combination of numbers, variables, and mathematical operations that can be simplified or evaluated. It is a fundamental concept in algebra, which is the branch of mathematics that deals with the study of symbols and the rules for manipulating them.
To translate the phrase "19 increased by twice a number" into an algebraic expression, we can use the following steps:
Identify the operation being performed: In this case, the phrase "increased by" indicates that we need to use the addition operator (+).
Identify the constants and variables: The constant in this expression is 19, and the variable is "a number", which we will represent with the variable n.
Write the expression: Using the information from steps 1 and 2, we can write the expression as follows:
19 + 2n
Hence, the expression represents the result of adding 19 to twice a number is 19 + 2n.
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The algebraic expression represents the result of adding 19 to twice a number is 19 + 2n.
What is an algebraic expression?
An algebraic expression is a combination of numbers, variables, and mathematical operations that can be simplified or evaluated. It is a fundamental concept in algebra, which is the branch of mathematics that deals with the study of symbols and the rules for manipulating them.
To translate the phrase "19 increased by twice a number" into an algebraic expression, we can use the following steps:
Identify the operation being performed: In this case, the phrase "increased by" indicates that we need to use the addition operator (+).
Identify the constants and variables: The constant in this expression is 19, and the variable is "a number", which we will represent with the variable n.
Write the expression: Using the information from steps 1 and 2, we can write the expression as follows:
19 + 2n
Hence, the algebraic expression represents the result of adding 19 to twice a number is 19 + 2n.
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You are making snack bags to sell at the fair. You want each bag to contain the same combination of carrot and celery sticks. You have 75 carrot sticks and 40 celery sticks and want to use them all.
What is the greatest number of snack bags you can make
The greatest number of snack bags is 10.
Highest Common Factor:The largest of all the common factors between two or more numbers is known as the Highest Common Factor (HCF). It is also known as the Greatest Common Element (GCF).
HCF is a fundamental method that allows you to equally distribute items throughout a group or set.
We can use the idea of HCF to determine the same when you are organizing a party and want to make sure that nothing is wasted or you require a proper calculation.
Here we have
Number of carrots = 75
Number of celery sticks = 40
Here we are making snack bags such that each bag contains the same combination of carrot and celery sticks.
To find the greatest number of snack bags we need to find HCF of 70 and 40 as given below
Write given numbers as the product of prime numbers
=> 70 = 2 × 5 × 7
=> 40 = 2 × 2 × 2 × 5
List out the common factors
=> 2 × 5
=> HCF (70, 40) = 10
Therefore,
The greatest number of snack bags is 10.
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A bowling ball is dropped from a height of 40
feet.
a. Write a function that gives the height h (in feet) of the bowling ball after t seconds. h(t) =?
b. Find the height of the bowling ball after 1 second. The height of the ball after 1 second is
feet.
c. How long does the bowling ball take to hit the ground? Round your answer to the nearest hundredth of a second. The bowling ball takes about _ seconds to hit the ground.
(•
The formula for glue says to add 55ml of hardener to each container of resin. How much hardener should be added to 18 containers of resin?
Write your answer in liters.
The quantity of the hardener added to 18 containers of resin will be 990 ml.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that the formula for glue says to add 55ml of hardener to each container of resin.
The quantity of hardener for 18 containers will be calculated as,
Q = 18 x 55
Q = 990 ml
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Select the correct answer.
A solution to a quadratic function, f, is given below.
-3-51
Which statement is true about the given function?
O
The other solution to function fis 3 - 5i.
The other solution to function fis 3 + 5i.
Function f has no other solutions.
The other solution to function fis-3 + 5i.
Answer:
(d) The other solution to function f is -3 + 5i
Step-by-step explanation:
Given a solution to quadratic function f is -3-5i, you want to know the other solution.
ConjugateThe complex solutions to quadratic equations come in conjugate pairs. That is, if one solution is -3-5i, the other solution will be the same thing with the sign of the imaginary part reversed:
The other solution is -3+5i.
__
Additional comment
This is true only if the quadratic has real coefficients. (We typically don't study any other kind.)
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A coach asked the frequency of people who bought football tickets, and the data is displayed in histogram
Which of the best describes the spread and distribution of data
The correct statement regarding the spread of the data from the frequency table is given as follows:
The data has an outlier in the 13 to 15 range and shows that a large group purchased tickets to the game.
How to classify the spread of data?To classify the spread of the data, we need to understand the frequency table, which shows that number of people who purchased each range of tickets.
The frequencies mean that:
12 people purchased between 1 and 3 tickets.16 people purchased between 4 and 6 tickets. (modal class, hence unimodal distribution).10 people purchased between 7 and 9 tickets.1 person purchased between 13 and 15 tickets.The person purchasing between 13 and 15 tickets represents an outlier, as the interval 13-15 is far from the other observations in the data-set.
This means that the first option gives the correct description of the spread of the data-set.
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Fully simplify. (3x^2y^2)^3
landon's ice cream machine produced 1.2 cups of ice cream in 3 minutes. If it produced the same amount each minute, how much ice cream did the machine produce during the 2nd minute?
The ice cream that the machine produces during the 2nd minute will be 0.8 cups of ice cream.
How to illustrate the rate?Rate demonstrates how many times one number can fit into another number. They contrast two numbers by ordinarily dividing them. A/B will be the formula if one is comparing one data point (A) to another data point (B).
Typically, a rate is defined as the ratio of two quantities expressed in different units. The first quantity is typically written as the numerator of a fraction, and the second quantity is written as the denominator. By condensing them into the simplest form possible, we can calculate the rate.
A rate is a ratio that contrasts two distinct quantities represented by various units. A comparison of two numbers with various quantities or units is called a rate. A ratio or a rate out of 100 is what is known as a percentage.
Here, landon's ice cream machine produced 1.2 cups of ice cream in 3 minutes. If it produced the same amount each minute, the ice cream that the machine produces during the 2nd minute will be:
= (1.2 / 3) × 2
= 0.4 × 2
= 0.8 cups.
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Using traditional methods, it takes 100 hours to receive a basic driving license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique with 260 students and observed that they had a mean of 99 hours. Assume the standard deviation is known to be 6. A level of significance of 0.01 will be used to determine if the technique performs differently than the traditional method. Find the value of the test statistic. Round your answer to 2 decimal places.
The value of the Test statistic =z=-1.67
We have the value of sigma = 6, n = 260
In a hypothesis test, a test statistic—a random variable—is computed from sample data. To decide whether to reject the null hypothesis, you can utilize test statistics. Your results are compared to what would be anticipated under the null hypothesis by the test statistic. The p-value is computed using the test statistic.
Now using the formula of Test static
Test statistic will be: [tex]\mathrm{z}=\frac{\overline{\mathrm{x}}-\mu}{\frac{\sigma}{\sqrt{\mathrm{n}}}}\\[/tex]
Where, n = total number of sample = 100
[tex]\bar x[/tex] = mean of the given data
[tex]\mu[/tex] = total number of hours taken to receive a basic driving licence.
So, putting the value given in the question,
We will get the value of test static as:
[tex]=\frac{99-100}{\frac{6}{\sqrt{260}}}=-1.67[/tex]
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While on a mountain hike, you see a sign posting that you are at an elevation of 130 ft You arrive at another sign 10
minutes later that says you are at an elevation of 240 ft
What was your average rate of change of elevation? Round your answer to the nearest whole number, if necessary.
Answer:
11 feet every minute
Step-by-step explanation:
We first need to figure out how far you traveled up.
240 - 130 = 110
Now, we can solve for the average rate of elevation.
If you travel 110 feet in 10 minutes, we can do this equation:
110 ÷ 10 = 11
The average change of elevation, every minute, was 11.
TRUE/ FALSE. A researcher compares the BMI of two populations of individuals from different socioeconomic backgrounds and finds that the proportion of individuals considered obese and from a poor background is .37 (n = 112), and the proportion of individuals considered obese and from an affluent background is .27 (n = 87); a 95% confidence interval for the risk difference for obesity among members of the poor population compared to the affluent population is (-0.03, 0.23).
True , a 95% confidence interval for the risk difference for obesity among members of the poor population compared to the affluent population is (-0.03, 0.23).
The critical value for alpha is 0.05 and z is 1.96 .
The corresponding confidence interval is ( 6.235 , 7.165).
Therefore the above statement is true.
Confidence Interval, to put it simply, is a range within which we are confident that the genuine value exists. The probability that a confidence interval will contain the real parameter value depends on the confidence level that is chosen for the interval.
The phrase "Confidence Interval" refers to the range of values that is typically used to handle population-based data, extracting precise, important information with a given level of confidence.
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Liana consumes only beer and chips. Her indifference curves are all bowed inward. Consider the bundles (2,6), (4,4), and (6,2). If Liana is indifferent between (2,6) and (6,2), then Liana must
prefer (4,4) to (6,2).
If Liana is indifferent between the bundles (2,6) and (6,2), then it means that she values these two bundles equally. This means that Liana does not have a preference for one bundle over the other.
If Liana's indifference curves are bowed inward, it means that she is willing to give up more units of one good in order to get more units of the other good, as she moves along the indifference curve. This implies that Liana has a diminishing marginal rate of substitution (MRS) between the two goods.
Therefore, if Liana is indifferent between (2,6) and (6,2), it does not necessarily follow that she prefers (4,4) to (6,2). It is possible that Liana may also be indifferent between (4,4) and (6,2), or she may prefer (6,2) to (4,4). This would depend on the specific shape and location of Liana's indifference curves.
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Calculate the average rate of change between x=-1 and x=1.
Between x=-1 and x=1, the average rate of change is 1.
What is meant by average rate of change?It represents the average change in the function's per-unit value during that time period. Calculating the average rate of change is very helpful for tracking changes in measurable parameters like average speed or average velocity. As the slope of the straight line that links the two spots, the average rate of change between the two points is determined. Use the formula:
(f(b)-f(a))/(b-a) to get the average rate of change of f(x) between x=a and x=b.
Given line in the figure passes through the points A(4, 0), B(-1, -5)
Let the equation of the line be:
y=mx+ c
Slope m=(-5-0)/(-1-4)
m=1
Therefore, y=x+ c
It passes through (4, 0)
y+ c=0
c=-4
y=x-4=f(x)
f(-1)= -1-4=-5
f(1)= 1-4=-3
The average rate of change=(f(b)-f(a))/(b-a)
=(f(1)-f(-1))/1-(-1)
= -3-(-5)/(1+1)
=2/2
=1
Therefore, Between x=-1 and x=1, the average rate of change is 1.
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The mass of a sample of bacteria has increased by 2% each day for the past 10 days. The initial mass was 5 mg. If this growth continues, which equation can be used to determine M, the mass after d days?
The growth of the mass of the sample of a bacteria is represented by the exponential model m(x) = 5 · 1.02ˣ.
How to predict the growth of the mass of the sample of a bacteria
In this problem we find the case of mass of a sample of a bacteria (m(x)), in miligrams growing in time (x), in days. This growth is done exponentially, the exponential model is shown below:
m(x) = m' · (1 + r / 100)ˣ
Where:
m' - Initial mass, in miligrams.r - Growth rate, in percentage.x - Time, in days.We have that the mass of the sample was initially 5 miligrams and grows at a rate of 2 % during 10 days. If we know that m' = 5 mg and r = 2, then the exponential model for the sample is:
m(x) = 5 · 1.02ˣ
The exponential model for the sample of a bacteria is equal to m(x) = 5 · 1.02ˣ.
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Use spherical coordinates to Find the average distance from a point in a ball of radius a to its center.
The average distance from a point in the ball to its center = [tex]\frac{3}{4}[/tex] A
Given the radius in the question = a
The integration of the total distance before dividing the volume will be =
⇒ [tex]\int_0^{2\pi}\int_0^\pi\int_0^A a\cdot a^2\sin\theta\,da\,d\theta\,d\phi[/tex]
⇒ [tex]\int_0^{2\pi}\int_0^\pi\int_0^A a^3\sin\theta\,da\,d\theta\,d\phi[/tex]
⇒ [tex]\frac{A^{4} }{2}[/tex] [tex]\int_0^{2\pi}\int_0^\pi \sin\theta\,d\theta\,d\phi[/tex]
⇒ [tex]\frac{A^{4} }{2}[/tex] [tex]\int_0^{2\pi}1\,d\phi[/tex]
⇒ [tex]\pi A^{4}[/tex]
Let us divide the average distance by the ball's volume,
⇒ [tex]\frac{\pi a^4}{(4/3)\pi a^3}[/tex]
⇒ [tex]{\frac{3a}4}[/tex]
By the use of the Parcly Taxel's way,
Let the radius of the ball be = A
f(x) = probability density function
Distance = D from the given point to the center.
P( D < a )
[tex]\begin{align}P(D\leq r)&= \frac{ \frac 43 \pi r^3 }{\frac 43 \pi R^3}=\frac {r^3}{R^3}, \ \ \ 0\leq r\leq R. \end{align}[/tex][tex]\frac{\frac{4}{3} \pi a^{3} }{\frac{4}{3} \pi A^{3} } = \frac{a^{3} }{R^{3} }[/tex]. 0 [tex]\leq[/tex] a [tex]\leq[/tex] A
Therefore, The average distance from a point in the ball to its center = [tex]\frac{3}{4}[/tex] A
To know more about Spherical coordinates,
https://brainly.com/question/19053603
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