8. for each of the given sample data sets below, calculate the mean, variance, and standard deviation. (a) 79, 52, 64, 99, 75, 48, 52, 24, 76 mean

Understanding the equilibria, sketching a phase portrait, and determining the stability of equilibria for autonomous equations are important tools for analyzing and understanding the behavior of systems over time.

Autonomous equations are differential equations that do not depend explicitly on time. To find the equilibria of an autonomous equation, we set the derivative of the function to zero and solve for the values of the independent variable that satisfy the equation. These values represent points at which the function does not change over time and are known as equilibrium points.

To sketch a phase portrait for an autonomous equation, we plot the slope field of the function and then draw solutions through each equilibrium point. The resulting graph shows the behavior of the function over time and helps us understand how the solutions behave near each equilibrium point.

The stability of an equilibrium point is determined by examining the behavior of nearby solutions. If nearby solutions move toward the equilibrium point over time, the equilibrium point is stable. If nearby solutions move away from the equilibrium point over time, the equilibrium point is unstable. Finally, if the behavior of nearby solutions is inconclusive, further analysis is needed.

Here is the sketch for [tex]dx/dt = x - x^3[/tex]

       / <--- (-∞)  x=-1  (+∞) ---> \

      /                                \

  <--0-->       x=-1       x=1        0-->

      \                                /

       \ <--- (-∞)  x=1   (+∞) --->  /

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The asymptotes of the function f(x) = (2x² - 5x + 3)/(x - 2) are given as follows:

The function for this problem is defined as follows:

f(x) = (2x² - 5x + 3)/(x - 2)

The vertical asymptote is the value of x for which the function is not defined, hence it is at the zero of the denominator, and thus it is given as follows:

x - 2 = 0

x = 2.

The oblique asymptote is at the quotient of the two functions, hence:

(mx + b)(x - 2) = 2x² - 5x + 3

mx² + (b - 2m) - 2b = 2x² - 5x + 3.

Hence the values of m and b are given as follows:

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Answer:

Step-by-step explanation:

Answer:

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&AB&BC&AC\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&3.61&4&3.61\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&4.47&5.39&3.61\\\cline{1-4}\end{array}[/tex]

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&m \angle A&m \angle B&m \angle C\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&67.38^{\circ}&56.31^{\circ}&56.31^{\circ}\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&82.87^{\circ}&41.63^{\circ}&55.49^{\circ}\\\cline{1-4}\end{array}[/tex]

Step-by-step explanation:

Place points A, B and C on the coordinate grid.

Alternatively, type the following into the input field as 3 separate inputs:

Triangle 1

Triangle 2

Use the Segment tool to join each pair of points.

Alternatively, type Segment( <Point>, <Point> ) into the input field (replacing <Point> with the letter name of the point) to create a segment between two points.

Record the length of each side.

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&AB&BC&AC\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&3.61&4&3.61\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&4.47&5.39&3.61\\\cline{1-4}\end{array}[/tex]

Use the Angle tool to measure each angle in the resulting triangle.

Alternatively, type Angle(Polygon(A, B, C)) into the input field to create all interior angles.

Record the measure of each angle.

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\sf Location&m \angle A&m \angle B&m \angle C\\\cline{1-4}\vphantom{\dfrac12} A(3,4),\;B(1,1),\;C(5,1)&67.38^{\circ}&56.31^{\circ}&56.31^{\circ}\\\cline{1-4}\vphantom{\dfrac12} A(4,5),\;B(2,1),\;C(7,3)&82.87^{\circ}&41.63^{\circ}&55.49^{\circ}\\\cline{1-4}\end{array}[/tex]

Note: All measurements have been given to the nearest hundredth (2 decimal places).

Using linear negative association, According to the  all four parts correct options are D ;A ;D ;D respectively

The slope of a line expresses a great deal about the linear relationship between two variables. If the slope is negative, there is a negative linear relationship, which means that as one variable increases, the other variable decreases. If the slope is zero, one increases while the other remains constant.

The first answer to the question is option D

The second answer to the question must be option A  

Option D must be chosen for the third question.

Option D  must be selected for Question 4.

Solution:

1.

square of 3 is 9

3 to the power of negative 2 is 1/ 9

cube of 3 is 27

3 to the negative power 3 is 1/27

2.

cylinder  volume =πr²h

Given value

pi =3.14

r=5

h=10

Volume=3.14×5²×10

cylinder volume =785m³

3.

When a point is rotated 90 degrees anticlockwise about the origin, it becomes the point (x,y) (-y,x).

The coordinates of Point N are (4, 3)

N' will be the new coordinates (-3, 4)

As a result, the y-coordinate of N' is 4.

4.

Option D must be selected for Question 4.

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The commission earned 4 percentage on the salesperson on the sale of furnaces is $1320.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we need to calculate a percentage of a number, we should divide it by its entirety and then multiply it by 100. The percentage therefore refers to a part per hundred. The word per cent means per 100. The letter "%" stands for it. The term "percentage" was adapted from the Latin word "per centum", which means "by the hundred". Percentages are fractions with 100 as the denominator.

by the question.

the commission that the salesperson earns on the sale of $33,000 worth of furnaces= 4% of 33,000 = 4× 330 = $1320

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Answer:

A and B are not mutually exclusive

Step-by-step explanation:

A and B are not mutually exclusive because P(A and B) > 0. If A and B were mutually exclusive, then they would have no outcomes in common and the probability of their intersection would be zero. However, in this case, they do share some outcomes, since P(A and B) is greater than zero.

Answer:

Step-by-step explanation:

To find the total length of both pieces of fabric in centimeters, we need to add the length of the first piece of fabric (142 cm) and the length of the second piece of fabric (2 meters).

However, we need to make sure that the units are consistent before we add the lengths. We can convert the length of the second piece of fabric from meters to centimeters by multiplying by 100. Therefore, the total length in centimeters is:

142 cm + 2 meters * 100 cm/meter = 142 cm + 200 cm = 342 cm

The option that correctly gives the answer is "Multiply 2 by 100, then add 142" (Option C).

Answer:

No

Step-by-step explanation:

Since a polygon has straight sides, with 3 or more, it cannot be a polygon since one side is curved.

Answer: 14,605 pieces.

Step-by-step explanation:

In the second quarter, the company produced 795 pieces more than in the first quarter.

So, the total pieces produced in the second quarter can be calculated as:

6905 + 795 = 7700

The total pieces produced in the first semester (two quarters) can be calculated as:

6905 + 7700 = 14,605

Therefore, the company produced 14,605 pieces in the first semester.

The number of pieces the company produced in the first semester was 14,605 pieces.

The question asks to calculate how many pieces a company produced in the first semester, considering the production of two quarters.

In the first quarter, the company produced 6,905 pieces, as indicated in the question.

Already in the second quarter, the company produced 795 more pieces than in the first quarter, which means that the production in the second quarter was:

6,905 + 795 = 7,700 pieces.

To know the company's total production in the first semester, just add the productions of the two quarters:

6,905 + 7,700 = 14,605 pieces

Answer:

Step-by-step explanation:

Number of cats = 8 - 3 = 5

Fraction that are cats [tex]=\frac{5}{8}[/tex]

[tex]\bold{Solution:}[/tex]

[tex]\Delta[/tex][tex]DE[/tex][tex]F[/tex] congruent to [tex]\Delta[/tex][tex]JKI[/tex]

[tex]\bold{FD=JI} \text{(corresponding angles of congruent triangles)}[/tex]

[tex]z + 22 = 3z[/tex]

[tex]\text{or,} \ z-3 z= -22[/tex]

[tex]\text{or,} \ -2z = -22[/tex]

[tex]\text{or,} \ z = \bold{11}[/tex]

[tex]\bold{EF=KI} \text{(corresponding angles of congruent triangles)}[/tex]

[tex]5y+13=6y[/tex]

[tex]\bold{y=13}[/tex]

Step-by-step explanation:

h(-2) = 25

h(-1) = 5

h(0) = 1

h(1) = 1/5

h(2) = 1/25

Answer:

[tex]96 \: {m}^{2} [/tex]

The correct answer is B

Step-by-step explanation:

First, we have to find the area of one side of the cube:

[tex]a(side) = 4 \times 4 = 16[/tex]

Now multiply this number by 6 (since the cube has 6 sides in total):

[tex]a(surface) = 16 \times 6 = 96[/tex]

Answer: B - 96 sq m

Step-by-step explanation:

The surface area is the area of all the squares added up. To find the area of one square, you multiply 4 x 4, which equals 16. Then, count the number of sides on the cube. There are 6 sides on this cube. So, you multiply        16 x 6. 96 is your total. And you can eliminate C because it says meters instead of square meters.

We can conclude that Maya's experimental probabilities fluctuate around the theoretical probability, but over a larger number of trials, the experimental probabilities should converge towards the theoretical probability.

What is probability?

Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

The theoretical probability of rolling a 5 on a fair die is 1/6, which means that if the die is rolled many times, we would expect to see a 5 about 1/6 of the time.

For the first 100 trials, Maya rolled a 5 on 25 of those trials. The experimental probability of rolling a 5 in this case is:

experimental probability = number of 5's rolled / number of trials

experimental probability = 25/100

experimental probability = 0.25

So, in the first 100 trials, Maya's experimental probability of rolling a 5 was 0.25.

For the first 200 trials, Maya rolled a 5 on 30 of those trials. The experimental probability of rolling a 5 in this case is:

experimental probability = number of 5's rolled / number of trials

experimental probability = 30/200

experimental probability = 0.15

So, in the first 200 trials, Maya's experimental probability of rolling a 5 was 0.15.

Comparing these experimental probabilities to the theoretical probability, we see that after 100 trials, Maya's experimental probability of rolling a 5 (0.25) is higher than the theoretical probability (1/6 ≈ 0.167). This suggests that Maya's sample of 100 trials was somewhat biased in favor of rolling a 5.

On the other hand, after 200 trials, Maya's experimental probability of rolling a 5 (0.15) is lower than the theoretical probability (1/6 ≈ 0.167). This suggests that Maya's sample of 200 trials was somewhat biased against rolling a 5.

Overall, we can conclude that Maya's experimental probabilities fluctuate around the theoretical probability, but over a larger number of trials, the experimental probabilities should converge towards the theoretical probability. This is known as the law of large numbers, which states that as the number of trials or observations increases, the experimental probability will tend to approach the theoretical probability.

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We might say that Maya's experimental probabilities oscillate about the theoretical probability, but after more trials, the experimental probabilities ought to converge to the theoretical probability.

What is probability?

Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes. Statistics is the study of events that follow a probability distribution.

A fair die has a theoretical probability of rolling a 5 of 1/6, therefore if the die is rolled several times, we can anticipate seeing a 5 roughly 1/6 of the time.

For the first 100 trials, Maya rolled a 5 on 25 of those trials. The experimental probability of rolling a 5 in this case is:

experimental probability = number of 5's rolled / number of trials

experimental probability = 25/100

experimental probability = 0.25

So, in the first 100 trials, Maya's experimental probability of rolling a 5 was 0.25.

For the first 200 trials, Maya rolled a 5 on 30 of those trials. The experimental probability of rolling a 5 in this case is:

experimental probability = number of 5's rolled / number of trials

experimental probability = 30/200

experimental probability = 0.15

So, in the first 200 trials, Maya's experimental probability of rolling a 5 was 0.15.

Comparing these experimental probabilities to the theoretical probability, we see that after 100 trials, Maya's experimental probability of rolling a 5 (0.25) is higher than the theoretical probability (1/6 ≈ 0.167). This suggests that Maya's sample of 100 trials was somewhat biased in favor of rolling a 5.

On the other hand, after 200 trials, Maya's experimental probability of rolling a 5 (0.15) is lower than the theoretical probability (1/6 ≈ 0.167). This suggests that Maya's sample of 200 trials was somewhat biased against rolling a 5.

Overall, we can conclude that Maya's experimental probabilities fluctuate around the theoretical probability, but over a larger number of trials, the experimental probabilities should converge towards the theoretical probability. This is known as the law of large numbers, which states that as the number of trials or observations increases, the experimental probability will tend to approach the theoretical probability.

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The distance  from Link to the Octorok  is 10.63 units.

We know that the distance between two points (x₁, y₁) and (x₂, y₂) is given by the formula below:

distance = √( (x₂ - x₁)² + (y₂ - y₁)²)

Here we want to find the distance from Link to the Octorok so Link can attack, so we need to get the distance between the points (-4, -5) and (3, 3).

The distance will be:

distance = √( (3 + 4)² + (3 + 5)²)

distance = √( (7)² + (8)²)

distance = √113

distance = 10.63

The distance is 10.63 units.

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Answer: he would be 2 standard deviations above the

Step-by-step explanation:

Answer:

hey baby

Step-by-step explanation:

hi thwrw honey i love you lol

The operation we use to simplify a ratio after finding the greatest common factor (GCF) is division.

Option A is the correct answer.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

To simplify a ratio after finding the greatest common factor (GCF), we use division.

We divide both terms of the ratio by the GCF.

This reduces the ratio to its simplest form.

Thus,

The operation we use to simplify a ratio after finding the greatest common factor (GCF) is division.

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Answer:

m = -1

Step-by-step explanation:

may not be accurate, I haven't done this in a while

Answer:

-1y−y1=m(x−x1)

y−6=−1(x+5)

y−6=−1x+(−1×5)

y−6=−1x+−5

y−6=−1x−5

y=−1x−5+6

y=−1x+1

y=−x+1

m=−1

b=1

Step-by-step explanation: Hope this helps!! Mark me brainliest!

The answer of the standard normal area for each of the following questions are given below respectively.

Standard normal area refers to the area under the standard normal distribution curve, which is a normal distribution with a mean of 0 and a standard deviation of 1.

a. P(1.24<Z<2.14) = 0.0912

b. P(2.03 <Z<3.03) = 0.0484

c. P(-2.03 <Z<2.03) = 0.9542

d. P(Z > 0.53) = 0.2977

Note: The standard normal distribution is a continuous probability distribution with mean 0 and standard deviation 1. The area under the curve represents probabilities and can be calculated using a standard normal distribution table or a calculator with a normal distribution function.

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Answer:

Step-by-step explanation:

y2-y1/x2-x1

-7-(-19)/-2-1

12/-2

-6

The slope is -6

For all values of x, y, and z, the expression 56x + 40y - 48z = 8(7x + 5y - 6z) holds true.

As an illustration, the phrase x + y is one where x and y are terms with an addition operator in between. There are two sorts of expressions in mathematics: numerical expressions, which only contain numbers, and algebraic expressions, which also include variables.

Indeed, for all values of x, y, and z, the expression 56x + 40y - 48z = 8(7x + 5y - 6z) holds true.

We can simplify both sides of the equation to understand why:

56x + 40y - 48z = 8(7x + 5y - 6z)

56x + 40y - 48z = 56x + 40y - 48z

As we can see, the equation is true for all values of x, y, and z because both sides are identical.

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The number of lionfish after 6 years will be 85,609. The recursive equation for [tex]f(n)[/tex] will be [tex]f(n) = 4242.42(1.65)^n - 1300n[/tex].

Consider the function:

[tex]y = a (1 \pm r)^x[/tex]

Where x is the number of times this growth/decay occurs, a = initial amount, and r = fraction by which this growth/decay occurs.

lionfish are considered an invasive species, with an annual growth rate of 65%.

Then the equation will be

[tex]f(n) = P(1.65)^n[/tex]

[tex]\text{P = initial population}[/tex]

A scientist estimates there are 7,000 lionfish in a certain bay after the first year.

[tex]7000 = P(1.65)[/tex]

    [tex]P = 4242.42[/tex]

Then the equation will be

[tex]f(n) = 4242.42(1.65)^n[/tex]

The number of lionfish after 6 years will be

[tex]f(n) = 4242.42(1.65)^6[/tex]

[tex]f(n) = 85608.58[/tex]

[tex]f(n) \cong 85,609[/tex]

If scientists remove 1,300 fish per year from the bay after the first year.

Then the recursive equation for f(n) will be

[tex]f(n) = 4242.42(1.65)^n - 1300n[/tex]

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Answer:

x= 13 =smaller number

Step-by-step explanation:

let x=smaller number

y=2x-6=larger number

x+y=45

x+2x-6=45 (equation)

3x=39

x=13

Answer:

[tex]x = 13[/tex] ..... smaller number

and

[tex]y = 32[/tex] ....... larger number

Step-by-step explanation:

Greetings!!!

Let the two numbers be X and Y

[tex]x + y = 45........ \:equation \: 1[/tex]

The larger number is y and is 6 less than twice the smaller number. which means:-

[tex](y - 6) = 2x........... \: equation \: 2[/tex]

so, now solve for y. from equation 2

[tex]y - 6 = 2x \\ y = 2x + 6[/tex]

Substitute equation 1 into equation 2

[tex]x + y = 45 \\ x + (2x + 6) = 45 \\ 3x + 6 = 45 \\ 3x = 45 - 6 \\ 3x = 39....divide \: both \: sides \: by \: 3 \\ x = 13[/tex]

To solve for y substitute x into the first equation

[tex](13) = + y = 45 \\ y = 45 - 13 \\ y = 32[/tex]

Finally, to be more sure make a cross check

[tex]y - 6 = 2x \\ 32 - 6 = 2(13) \\ 26 = 26[/tex]

If you have any questions tag it on comments

Hope it helps!!!

Answer:

H. 7

Step-by-step explanation:

Given x² + rx + 12 is equivalent to (x + 3)(x + s), equate the two expressions and expand the right side of the equation:

[tex]\begin{aligned}x^2+rx+12&=(x + 3)(x + s)\\ x^2+rx+12&=x^2 + sx + 3x + 3s\\x^2+rx+12&=x^2 + (s+3)x + 3s\end{aligned}[/tex]

To find the value of r, first find the value of s.

The constant term of the right-hand side must be equal to the constant term of the left-hand side. Therefore:

[tex]\implies 3s = 12[/tex]

Solve for s by dividing both sides of the equation by 3:

[tex]\implies s = 4[/tex]

Compare the coefficients of the terms in x:

[tex]\implies r = s + 3[/tex]

Substitute the value of s into the equation and solve for r:

[tex]\begin{aligned} \implies r &= s + 3\\&= 4 + 3\\&= 7\end{aligned}[/tex]

Therefore, the value of r is 7.

Answer:

[tex]\large\boxed{\sf r = 7 }[/tex]

Step-by-step explanation:

Correct question:- If r and s are constants and r² +rx + 12 is equivalent to (x+3)(x + s), what is the value of r?

Here we are given that , the expression (x+3)(x+s) is equal to r² + rx + 12 .

Firstly, expand the expression (x+3)(x+s) as ,

[tex]\implies (x+3)(x+s) \\[/tex]

[tex]\implies x(x+s)+3(x+s) \\[/tex]

[tex]\implies x^2 + xs + 3x + 3s \\[/tex]

Take out x as common,

[tex]\implies x^2 + (3+s)x + 3s \\[/tex]

Now according to the question,

[tex]\implies x^2 + (3+s)x + 3s = r^2 + rx + 12\\[/tex]

On comparing the respective terms , we get,

[tex]\implies r = 3 + s \\[/tex]

[tex]\implies 3s = 12 \\[/tex]

Solve the second equation to find out the value of s , so that we can substitute that in equation 1 to find "r" .

[tex]\implies 3s = 12 \\[/tex]

[tex]\implies s =\dfrac{12}{3}=\boxed{4} \\[/tex]

Now substitute this value in equation (1) as ,

[tex]\implies r = 3 + s \\[/tex]

[tex]\implies r = 3 + 4 \\[/tex]

[tex]\implies \underline{\underline{ \red{ r = 7 }}} \\[/tex]

and we are done!

The 99% confidence interval for the population proportion p is :lower bound= 0.225, upper bound 0.375

What does a confidence interval actually mean?

Your estimate's mean is added to and subtracted from by the estimate's range to create a confidence interval. If you repeat your test, within a specific level of confidence, this is the range of values you anticipate your estimate to fall within.

Given that,

n = 250

x = 75

Point estimate = sample proportion =  p = x / n = 75/250=0.3

1 -  p =1 - 0.3=0.7

At 99% confidence level the z is ,

α  = 1 - 99% = 1 - 0.99 = 0.0

α / 2 = 0.01 / 2 = 0.005

Z /2 = Z0.005 = 2.576  (  Using z table   )  

Margin of error = E = Zα / 2 * √(( p * (1 -  p)) / n)

                              = 2.576 (√((0.3*0.7) /250 )

                            E  = 0.075

A 99% confidence interval for population proportion p is ,

p - E < p <  p + E

0.3 -0.075 < p < 0.3 + 0.075

0.225< p < 0.375

The 99% confidence interval for the population proportion p is :lower bound= 0.225, upper bound 0.375.

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the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

How to find?

The given function is:

f(x) = 1 / (x² + 10x + 25)

To find the definite integral of this function over the interval [5, 7], we can use the following steps:

Rewrite the function using partial fraction decomposition:

f(x) = 1 / (x² + 10x + 25)

= 1 / [(x + 5)²]

Using partial fraction decomposition, we can write this as:

f(x) = A / (x + 5) + B / (x + 5)²

where A and B are constants to be determined. Multiplying both sides by the common denominator (x + 5)², we get:

1 = A(x + 5) + B

Setting x = -5, we get:

1 = B

Setting x = 0, we get:

1 = 5A + B

= 5A + 1

Solving for A, we get:

A = 0

Therefore, the partial fraction decomposition is:

f(x) = 1 / [(x + 5)²]

= 0 / (x + 5) + 1 / (x + 5)²

Use the formula for the definite integral of a power function:

∫ xⁿ dx = (1 / (n + 1))× x²(n + 1) + C

where C is the constant of integration.

Using this formula, we can find the antiderivative of the function 1 / (x + 5)²:

∫ 1 / (x + 5)² dx = -1 / (x + 5) + C

Evaluate the definite integral over the interval [5, 7]:

∫[5,7] 1 / (x + 5)² dx

= [-1 / (x + 5)] [from 5 to 7]

= (-1 / 12) - (-1 / 10)

= (-5 / 600)

Therefore, the definite integral of f(x) over the interval [5, 7] is (-5 / 600).

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Step-by-step explanation:

35:12

40:17

45:22

50:27

55:32

The derivative of the function z= x^2y+xy^2, x = 3t y = t^2 using the chain rule is given by dz/dt = 36t^3 + 15t^4.

Expressions are equals to,

z= x^2y+xy^2

x = 3t

y = t^2

Using the chain rule calculate dz/dt,

which states that if z is a function of x and y,

And x and y are both functions of t, then,

dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)

Using these expressions, calculate the value of dz/dt using the chain rule,

z= x^2y+xy^2

This implies,

dz/dx = 2xy + y^2

dz/dy = x^2 + 2xy

x = 3t

⇒ dx/dt = 3

y = t^2

⇒ dy/dt = 2t

Substituting these values into the chain rule formula, we get,

dz/dt = (2xy + y^2)(3) + (x^2 + 2xy)(2t)

        = [2(3t)(t^2 ) + (t^2)^2 ]3 + [(3t)^2 + 2(3t)(t^2)](2t )

        = [ 6t^3 + t^4 ]3 + [ 9t^2 + 6t^3 ]2t

        = 18t^3 + 3t^4 + 18t^3 + 12t^4

        = 36t^3 + 15t^4

Substituting the given expressions for x and y into z, we get,

z = (3t)^2(t^2) + (3t)(t^2)^2

   = 9t^4 + 3t^5

here also,

dz/dt = 36t^3 + 15t^4

Therefore, the value of the function using the chain rule dz/dt is equals to  36t^3 + 15t^4.

learn more about chain rule here

brainly.com/question/24391730

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