PLEASE HELPP also sorry the photo is sideways
Reflecting /\ (triangle) LMN across the horizontal line y = -1, we get its image /\ (triangle) L' M' N'. Suppose LL', MM', NN' intersect the line of reflection at S, T, and U as shown below.

Answer:

The value of x is 45

Step-by-step explanation:

The value of x is 45 degrees as per the concept of the polygon's interior angle.

To find the value of x in the irregular pentagon with interior angles measuring 90 degrees, 112 degrees, x degrees, (3x + 10) degrees, and 148 degrees, we can use the fact that the sum of the interior angles in any pentagon is 540 degrees.

Summing up the given interior angles, we have:

90 + 112 + x + (3x + 10) + 148 = 540

Combine like terms:

4x + 360 = 540

Subtract 360 from both sides:

4x = 180

Divide both sides by 4:

x = 45

Therefore, the value of x is 45 degrees.

To learn more about the interior angles;

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i’m petty sure you add them all up

Answer:

$2,500

Step-by-step explanation:

The situation states that the employees earn vacation pay at the rate of one day per month and their daily wage is $100. Also, it states that in july 25 employees qualify for one vacation day. So, in order to determine the amount of vacation benefit expense for july, you need to multiply the daily wage for the number of employees that got the benefit:

$100*25= $2,500

Answer:

50 teachers

Step-by-step explanation:

step 1

Find the number of teachers for a ratio of 1:12

1/12=x/2,400

x=2,400/12=200 teachers

step 2

Find the number of teachers for a ratio of 5:48

5/48=x/2,400

x=2,400*5/48=250 teachers

step 3

Find the difference

250-200=50 teachers

The number of teachers who joined the school is 50.

Step 1:

Let's denote the number of teachers initially as x and the number of students initially as 12x, based on the initial ratio of 1 teacher to 12 students.

So, initially, the total number of people in the school is x + 12x = 13x.

Step 2:

After some teachers join the school, the new ratio becomes 5 teachers to 48 students.

Now, the number of teachers is [tex]\(x + \text{number of teachers who joined}\)[/tex], and the number of students remains 12x.

Step 3:

So, the new total number of people in the school becomes [tex]\(x + \text{number of teachers who joined} + 12x\).[/tex]

According to the new ratio, [tex]\(\frac{x + \text{number of teachers who joined}}{12x} = \frac{5}{48}\)[/tex].

We can set up the equation:

[tex]\[\frac{x + \text{number of teachers who joined}}{12x} = \frac{5}{48}\][/tex]

Step 4:

Cross-multiply:

[tex]\[48(x + \text{number of teachers who joined}) = 5 \times 12x\][/tex]

Simplify:

[tex]\[48x + 48(\text{number of teachers who joined}) = 60x\][/tex]

[tex]\[48(\text{number of teachers who joined}) = 12x\][/tex]

Divide both sides by 48:

[tex]\[\text{number of teachers who joined} = \frac{12x}{48} = \frac{x}{4}\][/tex]

Step 5:

Given that there are initially 2400 students, we can set up another equation:

[tex]\[12x = 2400\][/tex]

Solve for x:

[tex]\[x = \frac{2400}{12} = 200\][/tex]

Now, plug in the value of x to find the number of teachers who joined:

[tex]\[\text{number of teachers who joined} = \frac{x}{4} = \frac{200}{4} = 50\][/tex]

Therefore, the number of teachers who joined the school is 50.

ANSWER

[tex]a_ {10} = \frac{25}{32} [/tex]

EXPLANATION

The given geometric sequence is

400, 200, 100...

The first term is

[tex]a_1=400[/tex]

The common ratio is

[tex]r = \frac{200}{400} = \frac{1}{2} [/tex]

The nth term is

[tex]a_n=a_1( {r}^{n - 1} )[/tex]

We substitute the known values to get;

[tex]a_n=400( \frac{1}{2} )^{n - 1} [/tex]

[tex]a_ {10} =400( \frac{1}{2} )^{10 - 1} [/tex]

[tex]a_ {10} =400( \frac{1}{2} )^{9} [/tex]

[tex]a_ {10} = \frac{25}{32} [/tex]

The amount Malia earns can be shown as an addition sentence by multiplying the price of the wristbands by the quantity sold. The sum of this addition is $10, which is what Malia earned on the first day of the fundraiser. This sum can be visualized on a number line with four equal jumps of $2.50 leading to a sum of $10.

a. The amount Malia earns can be represented as an addition sentence by multiplying the price of the wristbands ($2.50) by the number of wristbands sold (4). The addition sentence would look like this: $2.50 + $2.50 + $2.50 + $2.50.

b. The sum of the addition sentence above is $10. This means that Malia has earned $10 on the first day of the fundraiser by selling 4 wristbands at $2.50 each.

c. The sum on a number line can be shown by marking off four equal jumps of $2.50 starting from zero, which leads you to the total sum of $10 at the fourth jump.

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Answer:

• V = x(11 -2x)(16 -2x)

• x ≈ 2.1 in

Step-by-step explanation:

a) The volume of the box is the product of its depth (x), width (11 -2x) and length (16 -2x). The equation can be simply ...

v(x) = x(11 -2x)(16 -2x)

__

b) A graphing calculator can plot this equation directly. All that is needed is to enter it into the appropriate space provided by the calculator. The value of x that gives the greatest volume is the value that makes the function have a local maximum between x=0 and x=5.5 (where the volume is again zero).

That value of x is about 2.1 inches.

Answer:

The area is 154 cm²

Step-by-step explanation:

Since the formula for the area of a circle is pi times the radius squared, divide the diameter in half to get the radius (7). Then, square the radius (49). Next, multiply that by pi (153.938). After that round to the nearest whole number (154). Hope that helps!

-Kyra

Answer:

A = 154 cm^2

Step-by-step explanation:

We know the diameter of the circle

We need to find the radius

d = 2r

14 = 2r

Divide by 2

14/2 = 2r/2

7=r

Now we can use the formula for area

A = pi r^2

A = pi (7)^2

A = 49pi

Replace pi with 3.14

A = 49(3.14)

A = 153.83

Rounding to the nearest whole number

A = 154 cm^2

Answer:

x = 1 and y = 2

Step-by-step explanation:

Let apples are represented by x

and let oranges are represented by y

You purchase 5 pounds of apples and 2 pounds of oranges for $9. This line in equation format can be written as:

5x + 2y = 9

Your friend purchases 5 pounds of apples and 6 pounds of oranges for $17.

This line in equation format can be written as:

5x + 6y = 17

Now we have two equations:

5x + 2y = 9 -> eq (i)

5x + 6y = 17 -> eq(ii)

We can solve these equations to find the value of x and y.

Subtracting eq(i) from eq(ii)

5x + 6y = 17

5x + 2y = 9

-     -         -

_________

0+4y= 8

=> 4y = 8

y= 8/4

y = 2

Now, putting value of y in eq (i)

5x + 2y = 9

5x +2(2) = 9

5x +4 = 9

5x = 9-4

5x = 5

x = 1

so, x = 1 and y = 2

Answer:

174 cm²

Step-by-step explanation:

The figure is composed of a rectangle and a triangle, so

area of figure = area of rectangle + area of triangle

area of rectangle = 8 × 15 = 120 cm²

area of triangle = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the height )

here b = 12 and h = 15 - 6 = 9 cm

area of triangle = 0.5 × 12 × 9 = 6 × 9 = 54 cm²

Hence

area of figure = 120 + 54 = 174 cm²

Answer: y=12.287

Step-by-step explanation:

Answer:

y = 12.3 cm

Step-by-step explanation:

Using the cosine ratio in the right triangle to solve for y

cos35° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{y}{15}[/tex]

Multiply both sides by 15

15 × cos35° = y, thus

y ≈ 12.3 cm

The answer is:

The lowest term will be:

[tex]\frac{7}{8}[/tex]

Reducing a fraction to its lowest term means writing it its simplified form, so, performing the operation and simplifying we have:

[tex]\frac{1}{4}+\frac{5}{8}=\frac{(1*8)+(4*5)}{4*8}\\\\\frac{(1*8)+(4*5)}{4*8}=\frac{8+20}{32}=\frac{28}{32}[/tex]

Now, to reduce the fraction to its lowest term, we need to divide both numerator and denominator by a common number, for this case, it will be "4" since is the biggest whole number that both numerator and denominator can be divided by, so, we have:

[tex]\frac{\frac{28}{4} }{\frac{32}{4}}=\frac{7}{8}[/tex]

Hence, we have that the lowest term will be:

[tex]\frac{7}{8}[/tex]

Have a nice day!

Answer:

c

Step-by-step explanation:

Answer:

d

Step-by-step explanation:

Here the sum of 5+1 golf scores, divided by 6, must be 118:

120 + 112 + 130 + 128 + 124 + x

--------------------------------------------- = 118

                         6

Here, 120 + 112 + 130 + 128 + 124 + x is the total number of points needed to have an average of 118.  Answer d is the correct one.

Answer:

Jacob has golf scores of 120, 112, 130, 128, and 124.

He wants to have an average golf score of 118.

a. Find the sum of all the numbers in the problem, 120+112+130+128+124+118.

[tex]120+112+130+128+124+118[/tex]

= 732

b. Find the average score for the five golf games that Jacob has played.

[tex]\frac{120+112+130+128+124}{5}[/tex]

= 122.8

c. Determine the number of points that he needs in his next golf game.

Jacob will need a golf score of 94  in next game to achieve the average of 118.

Total score = [tex]120+112+130+128+124+x[/tex]

number of matches = 6

Average score = [tex]\frac{614+x}{6}=118[/tex]

[tex]614+x=708[/tex]

[tex]x=708-614[/tex]

x = 94

d. Determine how many total points are needed to have an average of 118.

Total points needed are [tex]614+94=708[/tex]

1.25 meters im pretty sure. i hope i helped

Answer:

I think the answer is "How much pineapple juice is needed to make 5 pitchers of punch?"

Step-by-step explanation:

I believe it's the second one, from 5 cups of juice, since you would divide 5 by 3 1/2

ANSWER

See explanation

EXPLANATION

The given triangle is a right angle triangle.

We use the mnemonics SOH CAH TOA.

The given angle is

[tex] \alpha [/tex]

The opposite is 8 units, the adjacent is 15 units and the hypotenuse is 17 units.

[tex] \sin( \alpha ) = \frac{opposite}{hypotenuse} = \frac{8}{17} [/tex]

[tex] \csc( \alpha ) = \frac{1}{ \sin( \alpha ) } = \frac{17}{8} [/tex]

[tex]\tan( \alpha ) = \frac{opposite}{adjacent} = \frac{8}{11} [/tex]

[tex] \cot( \alpha ) = \frac{1}{ \tan( \alpha ) } = \frac{15}{8} [/tex]

[tex]\cos( \alpha ) = \frac{adjacent}{hypotenuse} = \frac{15}{17} [/tex]

[tex] \sec( \alpha ) = \frac{1}{ \cos( \alpha ) } = \frac{17}{15} [/tex]

In a right-angled triangle ABC, the trigonometric functions based on the given sides equate to: sin(α) = Option 3, csc(α) = Option 4, cos(α) = Option 5, sec(α) = Option 2, tan(α) = Option 1, and cot(α) = Option 6.

In a right-angled triangle, the Trigonometric Functions can be calculated based on the sides of the triangle.

For your problem, in triangle ABC, given that AB = 15 (hypotenuse), BC = 8 (opposite α) and AC = 17 (adjacent to α), we can compute the values as follows:

sin(α) = opposite/hypotenuse = BC/AB = 8/15 = (Option 3)

csc(α) = 1/sin(α) = 15/8 = (Option 4)

cos(α) = adjacent/hypotenuse = AC/AB = 17/15 = (Option 5)

sec(α) = 1/cos(α) = 15/17 = (Option 2)

tan(α) = sin(α)/cos(α) = (BC/AB) / (AC/AB) = BC/AC = 8/17 = (Option 1)

cot(α) = 1/tan(α) = 17/8 = (Option 6)

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The probable question may be:

In Right angle triangle ABC, Angle ABC is 90 degree, Angle BAC is α , side AB= 15, side BC=8 and side AC=17.

please match the correct trigonometric function with the correct value.

sin (α) =

csc(α) =

tan(α) =

sec(α) =

cos(α)=

cot(α)=

Options:

1. 8/17

2. 15/17

3. 8/15

4. 15/8

5. 17/15

6. 17/8

Perimeter = 2w+2l. 2(6)+2l = 36 subtract 12 to get 2l=24 then divide by 2 so the length is 12 inches

The phase shift is 7.2 right

Answer:

The height of the rectangular prism is [tex]58.36\ m[/tex]

Step-by-step explanation:

we know that

The surface area of the rectangular prism is equal to

[tex]SA=2B+PH[/tex]

where

B is the area of the rectangular base

P is the perimeter of the rectangular base

H is the height of the prism

Find the area of the base B

[tex]B=14.2*15=213\ m^{2}[/tex]

Find the perimeter of the base P

[tex]P=2(14.2+15)=58.4\ m[/tex]

we have

[tex]SA=3,834\ m^{2}[/tex]

substitute and solve for H

[tex]SA=2B+PH[/tex]

[tex]3,834=2(213)+(58.4)H[/tex]

[tex]3,834=426+(58.4)H[/tex]

[tex]H=(3,834-426)/(58.4)[/tex]

[tex]H=58.36\ m[/tex]

Answer:

Step-by-step explanation:

You need the 6th row of Pascal's triangle which contains the numbers 1, 5, 10, 10, 5, 1

Fill in the expansion as follows, using those numbers and the fact that a = 1 and b = 1:

[tex]1(1a)^5(1b)^0+5(1a)^4(1b)^1+10(1a)^3(1b)^2+10(1a)^2(1b)^3+5(1a)^1(1b)^4+1(1a)^0(1b)^5[/tex]

That simplifies down nicely to

[tex]a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5[/tex]

Those are fun!  Pascal's triangle is one of the coolest things ever!

To expand (a+b)^5 using Pascal's Triangle, we can use the Binomial Theorem. The expansion can be written as (a^5) + 5(a^4)(b) + 10(a^3)(b^2) + 10(a^2)(b^3) + 5(a)(b^4) + (b^5).

To expand the binomial (a+b)^5 using Pascal's Triangle, we will use the Binomial Theorem. According to the theorem, the expansion of (a+b)^n can be written as:

(a+b)^n = (nC0)(a^n)(b^0) + (nC1)(a^(n-1))(b^1) + (nC2)(a^(n-2))(b^2) + ... + (nCn)(a^0)(b^n)

For (a+b)^5, the expansion would be:

(a+b)^5 = (5C0)(a^5)(b^0) + (5C1)(a^4)(b^1) + (5C2)(a^3)(b^2) + (5C3)(a^2)(b^3) + (5C4)(a^1)(b^4) + (5C5)(a^0)(b^5)

Simplifying further, we get:

(a+b)^5 = (a^5) + 5(a^4)(b) + 10(a^3)(b^2) + 10(a^2)(b^3) + 5(a)(b^4) + (b^5)

Answer:

(6 , -3)

Step-by-step explanation:

Given in the question,

point J(8,-6)

x1 = 8

y1 = -6

point L(-2,9)

x2 = -2

y2 = 9

Location of point K which is 1/5 of the way from J to L

which means ratio of point K from J to L is 1 : 4

a : b

1 : 4

xk = [tex]x1+\frac{a}{a+b}(x2-x1)[/tex]

yk = [tex]y1+\frac{a}{a+b}(y2-y1)[/tex]

Plug values in the equation

xk = 8 + (1)/(1+4) (-2-8)

xk = 6

yk = -6 (1)/(1+4)(9+6)

yk = -3

Answer:

what he said

Step-by-step explanation:

Answer:

Step-by-step explanation:

Question One

Multiply through by 2

2*1/2 * (2x + y ) = 21/2 * 2

Combine

2x + y = 21

Subtract 2x from both sides

y = 21 - 2x

Now equate the two given equations

y = 21 - 2x

y = 2x

Add 2x to both sides

2x = 21 - 2x

2x + 2x = 21

4x = 21

x = 21/4

x = 5 1/4

or

x = 5.25

Question 2

[tex]\dfrac{2x + 6}{(x + 2)^2} - \dfrac{2}{(x + 2)}[/tex]

multiply numerator and denominator of the second fraction by (x + 2)

[tex]\dfrac{2x + 6}{(x + 2)^2} - \dfrac{2(x + 2)}{(x + 2)*(x + 2)}[/tex]

Remove the numerator brackets in the right hand fraction. Look out for the minus sign.

[tex]\dfrac{2x + 6}{(x + 2)^2} - \dfrac{2(x + 2)}{(x + 2)^2}\\\\\dfrac{2x + 6- 2x - 4}{(x + 2)^2}}\\\\\dfrac{2}{(x + 2)^2}[/tex]

Answer:  [tex]\bold{x=\dfrac{21}{4}}[/tex]

Step-by-step explanation:

[tex]\dfrac{1}{2}(2x+y)=\dfrac{21}{2}\\\\\text{Multiply both sides by 2 to clear the denominator:}\\2x + y = 21\\\\\text{Now, the system is:}\bigg\{{2x+y=21\atop{y=2x}}\\\\\text{Substitute y in the first equation with 2x to solve for x:}\\2x + y = 21\\2x + 2x = 21\\.\qquad 4x=21\\\\.\qquad \large\boxed{x=\dfrac{21}{4}}[/tex]

Answer:  a = 2

Step-by-step explanation:

[tex].\quad \dfrac{2x+6}{(x+2)^2}-\dfrac{2}{x+2}\bigg(\dfrac{x+2}{x+2}\bigg)\\\\\\=\dfrac{2x+6}{(x+2)^2}+\dfrac{-2(x+2)}{(x+2)^2}\bigg\\\\\\\\=\dfrac{2x+6-2x-4}{(x+2)^2}\\\\\\=\dfrac{2}{(x+2)^2}\implies \large\boxed{a=2}[/tex]

Answer:

The answer is the letter B.

The first column represents the x-values, and the second row represents the y-values.

For that reason, if we have:

2x - 7y = -1

x + 3y = -5

Then, the matrix will be given by:

[ 2      -7

 1        3]

Then, the third colum will be the equality:

[ -1

 -5]

So the correct option is the letter B.

B

[tex]2x - 7y = - 1 \\ \\ \\ 1. \: 2x = - 1 + 7y \\ 2. \: 2x = 7y - 1 \\ 3. \: x = \frac{7y - 1}{2} [/tex]

the answer Is B) 3c(3ab+a+4b)

Answer

B) 3c(3ab+a+4ab)

Step-by-step explanation:

First find the common factor of (9abc+3ac+12bc) (the common factor is 3c because 3 is the greatest common factor of the coeffecients given and c is in all the terms of the variables given)

then, put 3c outside the parenthesis and factor the terms.

3c(3ab+a+4ab)

when you multiply 3c(3ab+a+4ab) you should get the polynomial that the question gave you. (9abc+3ac+12bc)

Answer:

  5

Step-by-step explanation:

Among the 20 digits shown, each digit appears in the list twice except 0 and 1 appear 3 times and 6 and 9 appear once. That means ...

So, if 1 and 2 represent red candies, there are 3+2 = 5 red candies in the simulated random sample of 20 candies.

_____

Comment on the question

The simulation makes sense only if it represents taking a single candy from each of 20 packages (of unknown quantity of candies). That is, it seems we cannot answer the question, "how many red candies will be in the packages?" We can only answer the question, "how many of the simulated candies are red?"

Let [tex]z=-\sqrt3+i[/tex]. Then

[tex]|z|=\sqrt{(-\sqrt3)^2+1^2}=2[/tex]

[tex]z[/tex] lies in the second quadrant, so

[tex]\arg z=\pi+\tan^{-1}\left(-\dfrac1{\sqrt3}\right)=\dfrac{5\pi}6[/tex]

So we have

[tex]z=2e^{i5\pi/6}[/tex]

and the fourth roots of [tex]z[/tex] are

[tex]2^{1/4}e^{i(5\pi/6+k\pi)/4}[/tex]

where [tex]k\in\{0,1,2,3\}[/tex]. In particular, they are

[tex]2^{1/4}e^{i(5\pi/6)/4}=2^{1/4}e^{i5\pi/24}[/tex]

[tex]2^{1/4}e^{i(5\pi/6+2\pi)/4}=2^{1/4}e^{i17\pi/24}[/tex]

[tex]2^{1/4}e^{i(5\pi/6+4\pi)/4}=2^{1/4}e^{i29\pi/24}[/tex]

[tex]2^{1/4}e^{i(5\pi/6+6\pi)/4}=2^{1/4}e^{i41\pi/24}[/tex]

Answer:

Step-by-step explanation:

1/2log8,342 is almost correct.  Should enclose that "1/2" inside parentheses.  The "1/2" stems from our needing to find the value of the square root of 8342.

Answer:  The required answer is [tex]\dfrac{1}{2}\log 8342.[/tex]

Step-by-step explanation:  We are given to use the logarithmic equation to find [tex]\sqrt{8342}.[/tex]

We will be using the following logarithmic property :

[tex]\log a^b=b\log a.[/tex]

Let us consider that

[tex]x=\sqrt{8342}~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

Applying logarithm on both sides of equation (i), we have

[tex]\log x=\log{\sqrt{8342}\\\\\Rightarrow \log x=\log(8342)^\frac{1}{2}\\\\\Rightarrow \log x=\dfrac{1}{2}\log 8342.[/tex]

Thus, the required answer is [tex]\dfrac{1}{2}\log 8342.[/tex]

Answer:

  4/r

Step-by-step explanation:

The side lengths s of an equilateral triangle inscribed in a circle of radius r will be ...

  s = r√3

The perimeter of the triangle will be 3s.

The area of the triangle will be s^2·(√3)/4.

Then the ratio P/A is ...

  P/A = (3s)/(s^2·(√3)/4) = (4√3)/s

Substituting the above expression for s, we have ...

  P/A = 4√3/(r√3)

  P/A = 4/r

The expression for the ratio of the perimeter to the area of an equilateral triangle, whose vertices lie on a circle with radius r, is 2√3/r.

The ratio of the perimeter to the area of an equilateral triangle is derived using the formulae related to the triangle and the circle on which it lies. Let's start with the formulas for the circumference of a circle C = 2πr, and the area of an equilateral triangle A = (√3/4)*s², where s is the side length of the triangle.

As the vertices of the triangle are on the circle, the side length s is equal to the diameter of the circle. Therefore, s = 2r. Also, the perimeter P = 3*s = 6r. Substituting the terms for A and P, we find that P/A = 6r/((√3/4)*(2r)²) = (24/√3)/4r = 6/√3r. This simplifies to 2√3/r after rationalizing the denominator.

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Answer:

  The number generator is fair. It picked the approximate percentage of red lollipops most of the time.

Step-by-step explanation:

The other answer choices represent various misinterpretations of the nature of the experiment or the meaning of the numbers generated.

___

A number generator can be quite fair, but give wildly varying percentages of red lollipops. Attached are the results of a series of nine (9) simulations of the type described in the problem statement. You can see that the symmetrical result shown in the problem statement is quite unusual. A number generator that gives results that are too ideal may not be sufficiently random.

Option D is correct as it indicates that the number generator picked a percentage of red lollipops close to the expected 80% most of the time, reflecting fairness and random sampling variability.

Let's analyze the question regarding the fairness of the number generator. We know that 80% of the lollipops in the bag are red.

Option A: The number generator picked red lollipops 90% of the time in 3 experiments. This slight variation could be due to sampling variability, not necessarily unfairness.

Option B: States the correct percentage of red lollipops was not chosen at all, which might suggest the generator is faulty, but without more data, it's inconclusive.

Option C: Indicating the generator picked red lollipops half the time is incorrect as it would be inconsistent with the given information.

Option D: The generator picked the approximate percentage of red lollipops most of the time. This is plausible as the results can vary slightly due to random sampling.

Based on this analysis, Option D seems to be the most accurate description, assuming the observed percentages are close to the theoretical 80% on average, reflecting a fair and random selection process.

To find the exact length of the curve defined by [tex]\( x = 9 + 9t^2 \) and \( y = 6 + 6t^3 \) for \( 0 \leq t \leq 4 \):[/tex]

1. Compute derivatives: [tex]\( \frac{dx}{dt} = 18t \) and \( \frac{dy}{dt} = 18t^2 \).[/tex]

2. Substitute into arc length formula:

[tex]\[L = \int_{0}^{4} \sqrt{(18t)^2 + (18t^2)^2} \, dt = \int_{0}^{4} 18t \sqrt{1 + t^2} \, dt\][/tex]

3. Use substitution [tex]\( u = 1 + t^2 \), \( du = 2t \, dt \):[/tex]

[tex]\[L = 9 \int_{1}^{17} \sqrt{u} \, du = 9 \left[ \frac{2}{3} u^{3/2} \right]_{1}^{17} = 6 (\sqrt{4913} - 1)\][/tex]

Final answer: The exact length of the curve is [tex]\( \boxed{6 (\sqrt{4913} - 1)} \).[/tex]

To find the exact length of the curve defined by the parametric equations [tex]\( x = 9 + 9t^2 \) and \( y = 6 + 6t^3 \) for \( 0 \leq t \leq 4 \),[/tex] we use the arc length formula for parametric curves:

[tex]\[L = \int_{a}^{b} \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt\][/tex]

Here, ( a = 0 ) and ( b = 4 ). First, we need to find the derivatives

Given [tex]\( x = 9 + 9t^2 \):[/tex]

[tex]\[\frac{dx}{dt} = \frac{d}{dt}(9 + 9t^2) = 18t\][/tex]

Given [tex]\( y = 6 + 6t^3 \):[/tex]

[tex]\[\frac{dy}{dt} = \frac{d}{dt}(6 + 6t^3) = 18t^2\][/tex]

Next, we substitute these derivatives into the arc length formula:

[tex]\[L = \int_{0}^{4} \sqrt{(18t)^2 + (18t^2)^2} \, dt\][/tex]

Simplify the expression inside the square root:

[tex]\[(18t)^2 + (18t^2)^2 = 324t^2 + 324t^4 = 324t^2 (1 + t^2)\][/tex]

Therefore, the integrand becomes:

[tex]\[L = \int_{0}^{4} \sqrt{324t^2 (1 + t^2)} \, dt = \int_{0}^{4} \sqrt{324} \sqrt{t^2 (1 + t^2)} \, dt\][/tex]

[tex]\[L = \int_{0}^{4} 18 \sqrt{t^2 (1 + t^2)} \, dt = \int_{0}^{4} 18 t \sqrt{1 + t^2} \, dt\][/tex]

We can simplify this integral by using the substitution[tex]\( u = 1 + t^2 \), hence \( du = 2t \, dt \). When \( t = 0 \), \( u = 1 \), and when \( t = 4 \), \( u = 17 \):[/tex]

[tex]\[L = 18 \int_{0}^{4} t \sqrt{1 + t^2} \, dt = 18 \int_{1}^{17} \sqrt{u} \cdot \frac{1}{2} \, du\][/tex]

[tex]\[L = 9 \int_{1}^{17} \sqrt{u} \, du = 9 \int_{1}^{17} u^{1/2} \, du\][/tex]

Integrate[tex]\( u^{1/2} \):[/tex]

[tex]\[\int u^{1/2} \, du = \frac{2}{3} u^{3/2}\][/tex]

Evaluate the definite integral:

[tex]\[L = 9 \left[ \frac{2}{3} u^{3/2} \right]_{1}^{17} = 9 \left( \frac{2}{3} \left[ 17^{3/2} - 1^{3/2} \right] \right)\][/tex]

[tex]\[L = 9 \cdot \frac{2}{3} \left( 17^{3/2} - 1 \right) = 6 \left( 17^{3/2} - 1 \right)\][/tex]

[tex]\[L = 6 \left( \sqrt{17^3} - 1 \right) = 6 \left( \sqrt{4913} - 1 \right)\][/tex]

Thus, the exact length of the curve is:

[tex]\[\boxed{6 (\sqrt{4913} - 1)}\][/tex]