3 A prism is completely filled with 96 cubes that have edge length of 1 2 cm. What is the volume of the prism? Enter your answer in the box. cm³

Answer:

P(X=3) = (1/11) = 0.09

P(X < 3) = (5/11) = 0.45

P(X > 3) = (5/11) = 0.45

Step-by-step explanation:

The complete, correct question is shown in the attached image to this solution.

From the graph and the table,

P(X=3) = (1/11) = 0.09 (as shown in the question)

The probability that this score shows a number less than 3 is p(X < 3)

P(X < 3) = P(X=1) + P(X=2)

= (3/11) + (2/11) = (5/11) = 0.45

The probability that this score shows a number greater than 3 is p(X > 3)

P(X < 3) = P(X=4) + P(X=5)

= (2/11) + (3/11) = (5/11) = 0.45

Hope this Helps!!!

Answer:

x=55/4

Step-by-step explanation:

Answer:

The central angle of the arc is 288 degrees

Step-by-step explanation:

The correct question is

A circle has a circumference of 8. It has an arc of length 32/5 . What is the central angle of the arc,in the degrees?

we know that

The circumference of the circle subtend a central angle of 360 degrees

so

uing a proportion

Find out the central angle for an arc of length 32/5

[tex]\frac{8}{360^o}=\frac{(32/5)}{x}\\\\x=360(32/5)/8\\\\x=288^o[/tex]

Answer:

288

Step-by-step explanation:

Answer:

96.24% probability that a SRS of 25 students will spend an average of between 600 and 700 dollars

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

[tex]\mu = 650, \sigma = 120, n = 25, s = \frac{120}{\sqrt{25}} = 24[/tex]

What is the probability that a SRS of 25 students will spend an average of between 600 and 700 dollars

This is the pvalue of Z when X = 700 subtracted by the pvalue of Z when X = 600. So

X = 700

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{700 - 650}{24}[/tex]

[tex]Z = 2.08[/tex]

[tex]Z = 2.08[/tex] has a pvalue of 0.9812

X = 600

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{600 - 650}{24}[/tex]

[tex]Z = -2.08[/tex]

[tex]Z = -2.08[/tex] has a pvalue of 0.0188

0.9812 - 0.0188 = 0.9624

96.24% probability that a SRS of 25 students will spend an average of between 600 and 700 dollars

Answer: 35 square feet

Step-by-step explanation:

Answer:

500

Step-by-step explanation:

Lets make the unknown number the letter a

0.9% can also be shown as 0.9/100

so 0.9 of a = 4.5

x(0.9/100) = 4.5

multiply 0.9 by x

(0.9×a)/100 = 4.5

multiply both sides by 100

0.9×a = 450

divide both sides by 0.9

a = 500

there you have it

To determine the number that 4.5 is 0.9% of, we convert 0.9% to a decimal by dividing by 100 and solve the resulting equation. The answer is that 4.5 is 0.9% of 500.

To find the number that 4.5 represents 0.9% of, we should first convert the percent to a decimal. According to B.4, a percent is converted to a decimal by dividing the percent value by 100. In our case, 0.9% becomes 0.009.

Once we have the decimal equivalent of the percent, we can set up the equation where 4.5 is 0.9% (0.009 in decimal form) of some number x:

4.5 = 0.009x

To solve for x, divide both sides of the equation by 0.009:

x = 4.5 / 0.009

x = 500

Therefore, 4.5 is 0.9% of 500.

Answer:

y=2

Step-by-step explanation: horizon shows where the dotted line crosses the y axis

The horizontal asymptotes is [tex]y=2[/tex]

Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.

From given figure it is observed that,

at [tex]y=2[/tex] the graph of the function approaches as x tends to +∞ or −∞.

Thus, the horizontal asymptotes is [tex]y=2[/tex]

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Given:

ΔABC undergoes a dilation by a scale factor and comes as ΔA'B'C'.

To show that both the triangles are similar.

Formula

If two triangles have three pairs of sides in the same ratio, then the triangles are similar.

[tex]Hypotenuse^2 = Base^2+Height^2[/tex]

Now,

In ΔABC,

AB = 18 unit

BC = 10 unit

So, [tex]AC^2 = AB^2+BC^2[/tex]

or, [tex]AC^2 = 18^2+10^2[/tex]

or, [tex]AC = \sqrt{424}[/tex]

Again,

In ΔA'B'C'

A'B' = 9 unit

B'C' = 5 unit

So, [tex]A'C' ^2 = A'B'^2+B'C'^2[/tex]

or, [tex]A'C'^2 = 9^2+5^2[/tex]

or, [tex]A'C' = \sqrt{106}[/tex]

Now,

[tex]\frac{AB}{A'B'} = \frac{18}{9} = 2[/tex]

[tex]\frac{BC}{B'C'} = \frac{10}{5} = 2[/tex]

[tex]\frac{AC}{A'C'} = \frac{\sqrt{424} }{\sqrt{106} } = 2[/tex]

Hence,

All the ratios are equal.

Therefore, we can conclude that,

ΔABC and ΔA'B'C' are similar.

Answer:

3(n + 4) ≤ 10

Answer:

the answer would be 2.....3•n+4=10

10-4=6

6÷3=2

n=2

Step-by-step explanation:

Answer:

Step-by-step explanation:

Hello!

The objective of this exercise is to compare the proportion of defective parts produced by machine 1 and machine 2.

The parameter of study is the difference between the population proportion of defective parts produced by machine 1 and the population proportion of defective parts produced by machine 2, symbolically: p₁ - p₂

The hypotheses are:

H₀: p₁ - p₂ ≤ 0

H₁: p₁ - p₂ > 0

α: 0.03

This hypothesis test is one-tailed to the right, which means that you will reject the null hypothesis with high values of the statistic.

To test the difference of proportions you have to use a standard normal distribution, the critical value will be:

[tex]Z_{1-\alpha }= Z_{1-0.03}= Z_{0.97}= 1.881[/tex]

The decision rule using the critical value approach is:

If [tex]Z_{H_0}[/tex] ≥ 1.881, the decision is to reject the null hypothesis.

If [tex]Z_{H_0}[/tex] < 1.881, the decision is to not reject the null hypothesis.

Considering the calculated [tex]Z_{H_0}[/tex]  < 1.881, the decision is to not reject the null hypothesis. Using a significance level of 3%, you can conclude that the difference between the population proportion of defective plastic parts produced by machine 1 and the population proportion of defective plastic parts produced by machine 2 is at most zero.

I hope this helps!

Answer:

There is evidence to conclude that both machines produce the same fraction of defective parts

Step-by-step explanation:

Attached is the solution

Answer: The test statistic is -3.322.

Step-by-step explanation:

Since we have given that

n = 431

[tex]\hat{p}=0.42[/tex]

Hypothesis would be :

[tex]H_0:p=0.5\\\\H_1:p\neq 0.5[/tex]

So, the test statistic would be

[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}\\\\z=\dfrac{0.42-0.5}{\sqrt{\dfrac{0.5\times 0.5}{431}}}\\\\z=\dfrac{-0.08}{0.02408}\\\\z=-3.322[/tex]

Hence, the test statistic is -3.322.

C.10

Explanation

From table critical value of U when n1 is 7 and n2 is 8 and symbol alpha is 0.05 then Ucrit=10

The Mann-Whitney U test was used to compare study times between Psychology and English majors. The U value computed was 0.5, which is less than the critical value of 46. Therefore, the correct answer is A) 07.

To determine if there is a difference in the studying time between Psychology and English majors, the student is using the Mann-Whitney U test. The Mann-Whitney U test is a non-parametric test used to compare differences between two independent groups when the dependent variable is either ordinal or continuous, but not normally distributed.

Here are the given study times:

First, we need to rank all the study times from both groups combined, from lowest to highest, and then sum the ranks for each group.

Ranks:
Psychology Majors: 16 (10.5), 12 (7), 13 (8), 10 (4), 9 (3), 10 (4), 8 (2)
English Majors: 10 (4), 25 (16), 15 (9), 17 (13), 23 (15), 14 (8), 19 (14), 18 (12)

Sum of ranks:
Psychology Majors: 49.5
English Majors: 91.5

Using these ranks and sums, we calculate the U values:

The Mann-Whitney U value is the smaller of U1 and U2, so U = 0.5. Given that Ucrit = 46 at α = 0.05, 2-tailed, we compare our U value to Ucrit.

Because 0.5 < 46, we reject the null hypothesis.

The correct answer choice is A) 07.

Answer:

a. Type I error (rejecting a null hypothesis when it is true).

b. Correct conclusion (reject a null hypothesis when it is false).

Step-by-step explanation:

The question is incomplete:

Now suppose that the results of carrying out the hypothesis test lead to rejection of the null hypothesis.

Classify that conclusion by error type or as a correct decision if in fact:

a. the mean post-work heart rate of casting workers equals the normal resting heart rate of 72 bpm.

b. exceeds the normal resting heart rate of 72 bpm.

In case a, the conclusion is wrong and we have rejected a null hypothesis that is true. This is a Type I error, and it has a probability equal to the level of significance α.

In case b, the conclusion is correct, as the mean post-work heart rate indeed exceeds the normal resting heart rate of 72 bpm.

Answer:

7,548

Step-by-step explanation:

Answer:

  7548

Step-by-step explanation:

7×1000 +4×10 +8×1 +5×100

  = 7000 +40 +8 +500

  = 7548

The values of c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a) are approximately ±√(7/3).

Based on the given information, the Mean Value Theorem can be applied to function f on the closed interval [a, b].

[tex]f(x) = 3x^3[/tex] is continuous on the closed interval [1, 2], and it is also differentiable in the open interval (1, 2).

To find the values of c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a), we need to find the derivative of f(x) and solve the equation.

The derivative of  [tex]f(x) = 3x^3[/tex] can be found by applying the power rule, which states that the derivative of [tex]x^n[/tex] is [tex]nx^{n-1}[/tex].

So, the derivative of  [tex]f(x) = 3x^3[/tex]  is [tex]f'(x) = 9x^2[/tex].

Now, we can solve the equation f'(c) = (f(b) - f(a))/(b - a) using the given values for a and b.

Plugging in a = 1 and b = 2, we have:

[tex]f'(c) = (f(2) - f(1))/(2 - 1)\\= (3(2)^3 - 3(1)^3)/(2 - 1)\\= (3(8) - 3(1))/(2 - 1)= (24 - 3)/(2 - 1)= 21/1= 21[/tex]

Setting f'(x) = 21, we have:

[tex]9x^2 = 21[/tex]

[tex]x^2 = 21/9\\x^2 = 7/3[/tex]

x = ±√(7/3)

Therefore, the values of c in the open interval (a, b) such that f'(c) = (f(b) - f(a))/(b - a) are approximately ±√(7/3).

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The Mean Value Theorem can be applied to the function f(x) = [tex]3x^3[/tex] on the interval [1,2] as it is continuous and differentiable on this interval. According to the theorem, a value 'c' exists in the interval (1, 2) that complies with the equation derived from the theorem.

The Mean Value Theorem can indeed be applied for the function f(x) = [tex]3x^3[/tex] on the interval [1,2]. Mean Value Theorem can be applied if the function satisfies two conditions: it has to be continuous on the closed interval [a, b], and it has to be differentiable on the open interval (a, b).

The function f(x) = [tex]3x^3[/tex] is both continuous and differentiable for all real numbers, which includes the interval from 1 to 2. Therefore, we can apply the Mean Value theorem.

According to the theorem, there exists a number 'c' in the open interval (a, b) such that f '(c) = (f(b) − f(a)) / (b − a). Our given function f(x) = [tex]3x^3[/tex] differentiates to f '(x) = 9x. Setting f '(c) equal to (f(b) − f(a)) / (b − a), we can solve for the value of 'c'.

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Answer:

2 a) In Randomized Block design there are two variables one is a blocking variable the other one will be the treatment variable. Here type of shoes is the treatment variable and the type of runner is the blocking variable. Blocking is the arrangement of subjects similar in certain characteristics in to a group. Here professional runners are different from recreational runners . Blocking is done to reduce variability within groups so that variability within blocks is less than the variability between blocks. Then, subjects within each block are randomly assigned one of the shoes.

b) Randomization is the process of assigning participants a specific treatment  so that each participant has an equal chance of being assigned a shoe A or B. Randomization is done using random number generation and assignment is made according to the random numbers The main purpose of randomization is to eliminate biases. If the person in a group are allowed to choose the shoes they may choose their preferred one based on their past experience of using it or one variety will be preferred by most of the subjects in the group spoiling the entire purpose of the study. for e.g a group of professionals coming from a particular region prefers type A . If randomization is employed in such a situation almost half of the professionals coming from a particular region gets type A and the other half may get type B thus eliminating the personal biases in choice. This way we can eliminate any possible biases that may arise in the experiment. So randomization and blocking are important for a randomized block design in order to minimize bias in the responses.

Step-by-step explanation:

Answer:

z = 1,2,3,4

Step-by-step explanation:

0³ + 1³ = 1³ = 1

0³ + 2³ = 2³ = 8

.

.

.

0³ + 4³ = 4³ = 64

Answer:

The variance of the measure of productivity = 141.67(to 2 d.p)

Step-by-step explanation:

The complete question and the step-by step explanation are contained in the files attached to this solution.

Answer:

Horizontal Stretch by Scale Factor 1/2.

Step-by-step explanation:

Consecutive even integers are like 2, 4, 6, etc. So the equation would be:

x + x+2 + x+4 + x+6 = 292

Combine the like terms, x and the number values, to get:

4x + 12 = 292

Isolate x by subtracting 12 on both sides:

4x = 280

Divide by 4:

x = 70

Then plug in the value for x:

70, 72, 74, 76

The four consecutive even integers that sum up to 292 are 70, 72, 74, and 76.

To find 4 consecutive even numbers that sum to 292, let’s first understand what consecutive even numbers are. Consecutive even numbers are even numbers that follow each other in order. For instance, 2, 4, 6, 8 are four consecutive even numbers because each number is 2 more than the previous number.

Let's label the first of our 4 consecutive even numbers as 'x'. Each subsequent number is an increase of 2, so we can label them as 'x+2', 'x+4', and 'x+6'.

According to the question, the sum of these four integers is 292, so we form an equation: x + (x+2) + (x+4) + (x+6) = 292. Combining like terms, we get 4x + 12 = 292. Subtracting 12 from both sides of the equation: 4x = 280. Dividing both sides by 4 to solve for x, we find that x = 70.

So, using our labels from earlier, our four integers are 70, 72, 74, and 76.

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Answer:

first  6 x 4 = 24

Step-by-step explanation:

second, 120 divided by by the hours which is 24.

so 120 divided by 24 = 5

5 = answer

Answer:

p(x = 3, λ = 5) = 0.14044

Step-by-step explanation:

Given

λ = 5 (the average number of tracks per square centimeter)

ε = 2.718 (constant value)

x = 3 (the variable that denotes the number of successes that we want to occur)

p(x,λ) = probability of x successes, when the average number of occurrences of them is λ

We can use the equation

p(x,λ) = λˣ*ε∧(-λ)/x!

⇒ p(x = 3, λ = 5) = (5)³*(2.718)⁻⁵/3!

⇒ p(x = 3, λ = 5) = 0.14044

Answer:

0.0108

Step-by-step explanation:

Let X denote the number of uranium fission tracks occurring on the average 5 per square centimetre.We need to find the probability that a 2cm² sample of this zircon will reveal at most three tracks. X follows Poisson distribution, λ = 5 and s = 2.

k = λs = 5×2 = 10

Since we need to reveal at most three tracks the required probability is:

P (X≤3) = P (X =0) + P (X =1) + P (X =2) + P (X =3)

P (X≤3)  = (((e^​-10) × (10)⁰)/0!) +  (((e^​-10) × (10)¹)/1! +  (((e^​-10) × (10)²)/2! + (((e^​-10) × (10)3)/3!

P (X≤3)  = 0.0004 + 0.0005 +0.0023 +0.0076

P (X≤3)  = 0.0108

Therefore, the probability that a 2cm² sample of this zircon will reveal at most three tracks is 0.0108

Answer:

81

54

Step-by-step explanation:

The pattern is that with every inscreasing number (2,3 etc) you add 27.

Answer:

a. discrete random variable

Step-by-step explanation:

A discrete random variable is one which may take on only a countable number of distinct values such as 0, 1, 2, 3, 4, etc. Discrete random variables are usually countable.

From the question:

The experiment consist of making 80 telephone calls in order to sell a particular insurance policy. 80 here is the random variable; 80 is countable anf finite.

So, 80 is a discrete random variable

In the experiment involving making 80 telephone calls to sell an insurance policy, the random variable is the outcome of each call, which can be represented as a countable number. Therefore, it's a discrete random variable, where possibilities could be represented by integers, unlike a continuous random variable which can take on infinite possible outcomes.

In the context of the question, where an experiment involves making 80 telephone calls to sell a particular insurance policy, the random variable is defined as the outcome of each call — specifically, whether each call results in a sale or not. This outcome is countable and finite, so it is a discrete random variable.

By definition, a discrete random variable is a variable that can only take on a finite or countable number of values. Some examples of discrete random variables are the number of books on a shelf or the number of students in a class. Here, in the case of the telephone calls, the possibilities could be represented by integers (e.g., 0 indicating no sale and 1 indicating a sale).

On the other hand, a continuous random variable, such as the weight of a book or the amount of time a telephone call lasts, can take on any value within a specified range. These variables are associated with measurements and can have infinite possible outcomes within a given interval.

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Answer:

True

Step-by-step explanation:

The estimate of the true proportion of park users who are residents outside of the city is the number of park users in the sample who are residents outside of the city divided by the size of the sample.

In this problem:

26 park users resided outside the city, in a sample of 200 park users.

So

p = 26/200 = 0.13

So the answer is True

Answer:

[tex]t=\frac{4.8-4.5}{\frac{2.2}{\sqrt{81}}}=1.227[/tex]    

[tex]p_v =P(t_{(80)}>1.227)=0.1117[/tex]  

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to fail reject the null hypothesis, so we can't conclude that the true mean is higher than 4,5 years at 1% of signficance.  

Step-by-step explanation:

Data given and notation  

[tex]\bar X=4.8[/tex] represent the sample mean

[tex]s=2.2[/tex] represent the sample standard deviation

[tex]n=81[/tex] sample size  

[tex]\mu_o =4.5[/tex] represent the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

[tex]p_v[/tex] represent the p value for the test (variable of interest)  

State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean is higher than 4.5, the system of hypothesis would be:  

Null hypothesis:[tex]\mu \leq 4.5[/tex]  

Alternative hypothesis:[tex]\mu > 4.5[/tex]  

If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex]  (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Calculate the statistic

We can replace in formula (1) the info given like this:  

[tex]t=\frac{4.8-4.5}{\frac{2.2}{\sqrt{81}}}=1.227[/tex]    

P-value

The first step is calculate the degrees of freedom, on this case:  

[tex]df=n-1=81-1=80[/tex]  

Since is a one side test the p value would be:  

[tex]p_v =P(t_{(80)}>1.227)=0.1117[/tex]  

Conclusion  

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to fail reject the null hypothesis, so we can't conclude that the true mean is higher than 4,5 years at 5% of signficance.  

There is not enough evidence to support the claim that the mean time for all college students to earn their bachelor's degrees is greater than 4.5 years at the 0.05 significance level.




To test the claim that the mean time for all college students to earn their bachelor's degrees is greater than 4.5 years using a 0.05 significance level, follow these steps:
1. State the hypotheses:
  - Null hypothesis (H0): The mean time to earn a bachelor's degree is 4.5 years or less (mu <= 4.5).
  - Alternative hypothesis (H1): The mean time to earn a bachelor's degree is greater than 4.5 years (mu > 4.5).
2. Ditermine the test statistic:
  Since the sample size is large (n = 81), we use the z-test. The test statistic for the mean is calculated using the formula:
  z = (x - mu) / (s / sqrt(n))
  Where:
  - x is the sample mean (4.8 years)
  - mu is the population mean under the null hypothesis (4.5 years)
  - s is the sample standard deviation (2.2 years)
  - n is the sample size (81)
  Calculate the test statistic:
  z = (4.8 - 4.5) / (2.2 / sqrt(81))
  z = 0.3 / (2.2 / 9)
  z = 0.3 / 0.2444
  z ≈ 1.23
3. Find the p-value:
  Since this is a one-tailed test (greater than), we look up the cumulative probability for z = 1.23 in the standard normal distribution table or use a calculator.
  The cumulative probability for z = 1.23 is approximately 0.8907.
  The p-value is:
  p-value = 1 - 0.8907
  p-value = 0.1093
4.Compare the p-value to the significance level (alpha):
  The significance level is 0.05.
  If the p-value is less than alpha, we reject the null hypothesis.
  In this case:
  p-value = 0.1093
  alpha = 0.05
  Since 0.1093 > 0.05, we do not reject the null hypothesis.

5.Conclusion:
 There is not enough evidence to support the claim that the mean time for all college students to earn their bachelor's degrees is greater than 4.5 years at the 0.05 significance level.

Answer:

a) (1 + x + x^2 + x^3) ^5

b) (x^3 + x^4 + x^5 +x^6) ^3

c) ( x + x^2 + x^3 + x^4..........) ^6

d) ( 1 + x + x^2 + x^3 + x^4 + x^5) ^3

Step-by-step explanation:

A generating function is a process of encoding an infinite sequence of numbers (ar) by giving them a treatment as the coefficients of a power series. This formal power series is the generating function. As opposed to an ordinary series, this formal series is allowed to diverge, implying that the generating function is not always a true function and the "variable" is typically an indeterminate.

From the information above, build a generating function for ar, the number of distribution of r identical objects into:

(a) 5 different boxes with at most three objects in each boxes, this would be done as follows:

Answer = (1 + x + x^2 + x^3) ^5

(b) Three different boxes with between three and six objects in each boxes.

The answer is:

Answer= (x^3 + x^4 + x^5 +x^6) ^3

(c) Six different boxes with at least one object in each box.

The answer is:

Answer= ( x + x^2 + x^3 + x^4..........) ^6

(d) Three different boxes with at most five objects

The answer is:

Answer = ( 1 + x + x^2 + x^3 + x^4 + x^5) ^3

Answer:

a) ar = ( 1 + x + x^2 + x^3)^5

b) ar = ( x^3 + x^4 + x^5 + x^6 )^3

c) ar = ( x^1 + x^2 + x^3 + x^4 + ....)^6

d) ar = ( 5 + x^1 + x^2 + x^3 + x^4 + x^5 )^3

Step-by-step explanation:

Solution:-

- The generating function (ar), where the number of (r) identical objects.

- The number of identical boxes  = r

- The function parameter, the number of different boxes = n

- The number of objects in each box = k

- The general generating function (ar) is of the form:

               ar = (x^0 + x^1 + x^2 + x^3 + x^4 + ....+ x^k)^n

part a)

- We have 5 different boxes, n = 5.

- We are to place at most 3 objects in each box, k ≤ 3

- The generating function would be:

             ar = ( 1 + x + x^2 + x^3)^5

part b)

- We have 3 different boxes, n = 3.

- We are to place 3 to 6 objects in each box, (3 ≤ k ≤ 6)

- The generating function would be:

             ar = ( x^3 + x^4 + x^5 + x^6 )^3

part c)

- We have 6 different boxes, n = 6.

- We are place at-least 1 objects in each box, k ≥ 1

- The generating function would be:

             ar = ( x^1 + x^2 + x^3 + x^4 + ....)^6

part d)

- We have 3 different boxes, n = 3.

- We are place at-most 5 objects in each box, k ≤ 5

- The generating function would be:

             ar = ( 5 + x^1 + x^2 + x^3 + x^4 + x^5 )^3

The diameter of the traffic cone= 0.4604

Step-by-step explanation:

The volume of the cone= 150 cubic inches.

The height of the cone= 18 inches.

The volume of the cone= (1/3) x pi x R² x H

                                150 = (1/3) x (22/7) x R² x 18

                                  R² = 3 x 7/ 22 x 18

                                  R² = 0.0530

                                   R = 0.2302

                   Diameter, D= R x 2

                                    D= 0.2302 x 2

                   Diameter, D= 0.4604