Area of the Region Between Curves-Please help.

Answer:

4 months

Step-by-step explanation:

Club A

registration fee: $0

monthly fee: $105

After every month, the total cost increases by $105.

month 0: $0

month 1: $105

month 2: $210

month 3: $315

month 4: $420

month 5: $525

month 6: $630

Club B

registration fee: $375

monthly fee: $80

Notice how Club B's total reaches Club A's total after 2 months.

month 0: $375

month 1: $455

month 2: $535

month 3: $615

month 4: $695

Answer:

Pay for the day = $ 123.25

Step-by-step explanation:

From the question given:

Monday morning:

Time in: 8:15

Time out: 12:15 pm

Monday afternoon:

Time in: 13:00

Time out: 17:30

Pay = $ 14.5 /hr

Next, we shall determine the number of hours of work in the morning. This is illustrated below:

Time in: 8:15

Time out: 12:15 pm

Difference in time = 12:15 – 8:15 = 4 hrs

Next, we shall determine the pay for the work done in the morning. This can be obtained as follow:

Pay = $ 14.5 /hr

Pay for work done in the morning

= 4 × 14.5 = $ 58

Next, we shall determine the number of hours of work in the afternoon. This is illustrated below:

Time in: 13:00

Time out: 17:30 pm

Difference in time = 17:30 – 13:00 = 4 hrs 30 minutes

Next, we shall convert 4 hrs 30 minutes to hours. This is illustrated below:

60 minutes = 1 hr

30 minutes = 30/60 = 0.5 hrs.

Therefore,

4 hrs 30 minutes = 4 + 0.5 = 4.5 hrs

Next, we shall determine the pay for the work done in the afternoon. This can be obtained as follow:

Pay = $ 14.5 /hr

Pay for work done in the afternoon

= 4.5 × 14.5 = $ 65.25

Finally, we shall determine the pay for the day as follow:

Pay for work done in the morning

= $ 58

Pay for work done in the afternoon

= $ 65.25

Pay for the day = pay for morning + pay for afternoon

Pay for the day = $ 58 + $ 65.25

Pay for the day = $ 123.25

Therefore, the pay for the day is

$ 123.25

Answer:

[tex] \frac{7}{3} [/tex]

Step-by-step explanation:

Given that,

p = -6,

q = 6

r = -19

Plug in the above values to evaluate the expression, [tex] \frac{\frac{q}{2} - \frac{r}{3}}{\frac{3p}{6} + \frac{q}{6}} [/tex]

[tex] \frac{\frac{6}{2} - \frac{(-19)}{3}}{\frac{3(-6)}{6} + \frac{6}{6}} [/tex]

[tex] \frac{\frac{3}{1} - \frac{(-19)}{3}}{\frac{-3}{1} + \frac{1}{1}} [/tex]

[tex] \frac{\frac{9 -(-19)}{3}}{3 + 1} [/tex]

[tex] \frac{\frac{28}{3}}{4} [/tex]

[tex] \frac{28}{3}*\frac{1}{4} [/tex]

[tex] \frac{28*1}{3*4} [/tex]

[tex] \frac{7*1}{3*1} [/tex]

[tex] \frac{7}{3} [/tex]

Answer:

term=14x+19

cofficient=14+19

constant=19

Answer:

Terms :

14x , 19

Coefficients:

14

Constants:

19

Hope it helps.

Answer:

{-3, 2}U{2, 5}

Step-by-step explanation:

For an equation to be negative, it would need to be in a negative range (below the x-axis or the coordinates are negative y-values). Therefore, we can examine this question and see that the graph is negative when the function crosses the x-axis at -3 and it remains negative until you reach 2 on the x-axis.

Therefore, the first set of negative values is (-3, 2).

Secondly, applying the same logic as before, the function decreases at 2 and then touches the x-axis again at 5. Therefore, the second negative value would be (2, 5).

The negative values are {-3, 2}U{2, 5}.

Answer:

{-3, 2}U{2, 5}

Step-by-step explanation:

Answer:

this is wind turbine angular speed

Step-by-step explanation:

given data

angular speed ω = 64 rpm

time = 1 min = 60 seconds

solution

we know that angular speed ω is expess as

ω = [tex]\frac{2\pi }{T}[/tex]    .........................1

ω = 64 × [tex]\frac{2\pi }{T}[/tex]

ω  = 6.70 rad/s

so this is wind turbine angular speed

Answer:

A) A = 200w - w²

B) w = 100 yards

C) Max Area = 10000 sq.yards

Step-by-step explanation:

We are told that Robert has available 400 yards of fencing.

A) we want to find the expression of the area in terms of the width "w".

Since width is "w", and perimeter is 400,if we assume that length is l, then we have;

2(l + w) = 400

Divide both sides by 2 gives;

l + w = 200

l = 200 - w

Thus, Area of rectangle can be written as;

A = w(200 - w)

A = 200w - w²

B) To find the value of w for which the area is largest, we will differentiate the expression for the area and equate to zero.

Thus;

dA/dw = 200 - 2w

Equating to zero;

200 - 2w = 0

2w = 200

w = 200/2

w = 100 yards

C) Maximum area will occur at w = 100.

Thus;

A_max = 200(100) - 100(100)

A_max = 10000 sq.yards

Answer:

A. 2880 acres

Step-by-step explanation:

Formula: a = 640 s

Given information: s = 4.5

--> a = 640 x 4.5 = 2880 (acres)

Answer:

Well, there is a lot of answers to that. Check explanation, please!

Step-by-step explanation:

For example, some basic fractions that are less than 1/7 are:

You can also set up a number line, and compare the fractions.

Hopefully, this answer helps! :D

Answer:

Remainder = (3145/8)x - 408

Step-by-step explanation:

We want to find the remainder when 6x^(5) + 4x⁴ - 27x³ - 7x² + 27x + 3/2 is divided by (2x² - 3)²

Let's expand (2x² - 3)² to give ;

(2x - 3)(2x - 3) = 4x² - 6x - 6x + 9 = 4x² - 12x + 9

So,we can divide now;

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

First of all, we'll divide the term with the highest power inside the long division symbol by the term with the highest power outside the division symbol. This will give;

3/2x³

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

We now subtract the new multiplied term beneath the original one from the original one to get;

3/2x³

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

22x⁴+(27/2)x³+27x +3/2

We'll now divide the term in new polynomial gotten with the highest power by the term with the highest power outside the division symbol. This gives;

(3/2)x³ + (11/2)x²

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

22x⁴+(27/2)x³+27x +3/2

22x⁴-66x³ + (99/2)x²

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

We now subtract the new multiplied term beneath the immediate one from the immediate one to get;

(3/2)x³ + (11/2)x²

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

22x⁴+(27/2)x³-7x²+27x +3/2

22x⁴-66x³ + (99/2)x²

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(159/2)x³-(113/2)x²+27x+(3/2)

We'll now divide the term in new polynomial gotten with the highest power by the term with the highest power outside the division symbol. This gives;

(3/2)x³ + (11/2)x² + (159/8)x

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

22x⁴+(27/2)x³-7x²+27x +3/2

22x⁴-66x³ + (99/2)x²

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(159/2)x³-(113/2)x²+27x+(3/2)

(159/2)x³-(477/2)x²+(1431/8)x

We now subtract the new multiplied term beneath the immediate one from the immediate one to get;

(3/2)x³ + (11/2)x² + (159/8)x

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

22x⁴+(27/2)x³-7x²+27x +3/2

22x⁴-66x³ + (99/2)x²

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(159/2)x³-(113/2)x²+27x+(3/2)

(159/2)x³-(477/2)x²+(1431/8)x

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

182x²-(1215/8)x + (3/2)

We'll now divide the term in new polynomial gotten with the highest power by the term with the highest power outside the division symbol. This gives;

(3/2)x³+(11/2)x²+(159/8)x+(91/2)

______________________

4x²-12x+9 |6x^(5)+4x⁴-27x³-7x²+27x+3/2

6x^(5)-18x⁴-(27/2)x³

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

22x⁴+(27/2)x³-7x²+27x +3/2

22x⁴-66x³ + (99/2)x²

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

(159/2)x³-(113/2)x²+27x+(3/2)

(159/2)x³-(477/2)x²+(1431/8)x

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

182x²-(1215/8)x + (3/2)

182x²-545x + 819/2

We now subtract the new multiplied term beneath the immediate one from the immediate one to get;

182x² - (1215/8)x + (3/2) - 182x² + 545x - 819/2 = (3145/8)x - 408

Remainder = (3145/8)x - 408

Answer:

The slope would be 7/3

The y-intercept would be -11

Step-by-step explanation:

To find the answer rearrange your equation.

First subtract 14x from both sides, getting -6y= -14x + 66.

Then divide both sides by -6, getting y= 7/3x -11

This is now written in y=mx+b form. Your m is your slope and b is the y intercept!

Answer:

the answer is 10.41. If you have a number 5 or more you round the nearest number on the left up 1 if it's 4 or less it stays the same it doesn't go up or down.

Answer:

The answer is below

Step-by-step explanation:

Velocity is the rate of change of displacement. Velocity is the ratio of distance to time.

The velocity v(t) = [tex]\frac{d}{dt}r(t)[/tex]

Where r(t) is the position function

Given that:

r(t)=⟨−8tsint,−8tcost,2t²⟩

[tex]v(t)=\frac{d}{dt}r(t)= <-8tcost-8sint,8tsint-8cost,4t>[/tex]

Acceleration is the rate of change of velocity, it is the ratio of velocity to time. Acceleration a(t) is given as:

[tex]a(t)=\frac{d}{dt}v(t)= \frac{d}{dt} <-8tcost-8sint,8tsint-8cost,4t>\\=<8tsint-16cost,8tcost+16cost,4>\\\\a(t)=<8tsint-16cost,8tcost+16cost,4>[/tex]

Speed = |v(t)| = [tex]\sqrt{(-8tcost-8sint)^2+(8tsint-8cost)^2+(4t)^2}\\\\ =\sqrt{64t^2cos^2t+128tcostsint+64sin^2t+64t^2sin^2t-128tsintcost+64cos^2t+16t^2}\\ \\=\sqrt{64t^2cos^2t+64t^2sin^2t+64sin^2t+64cos^2t+16t^2}\\\\=\sqrt{64t^2(cos^2t+sin^2t)+64(sin^2t+cos^2t)+16t^2}\\\\=\sqrt{64t^2+64+16t^2}=\sqrt{80t^2+64}[/tex]

Answers:

Distance = 14 km

Displacement = 4 km east

==============================================

Explanation:

The total distance is simply the sum of the values given. So he traveled a total of 5+4+5 = 9+5 = 14 km overall.

----------------------

The displacement is where a drawing may come in handy. Draw an xy axis. Place point A at the origin (0,0). This is where Michael starts his trip. Then move up to (0,5) to indicate he goes 5 km north. Call this point B.

Afterward, mark point C at (4,5) to show he has gone 4 km east. Finally, point D is at (4,0) because he went 5 km south

The displacement is only concerned with two points: The start and end point. Nothing else matters. We started at A(0,0) and ended at D(4,0). So we've gone 4 km east overall. The direction is important when it comes to displacement. Simply saying "4 km" isn't enough. So in a sense, we could take a shortcut from A directly to D, bypassing the other points B and C.

See the diagram below for a visual.

Answer:

P ([tex]\hat p[/tex] ≤ 0.10)

Step-by-step explanation:

The probability in terms of statistics for this given problem can be written as follows.

Let consider X to the random variable that represents the number of 6's in 7 throws of a dice, then:

X [tex]\sim[/tex]Bin ( n = 72, p = 0.167)

E(X) = np

E(X) = 72× 0.167

E(X) = 12.024

E(X) [tex]\simeq[/tex]  12

p+q =1

q = 1 - p

q = 1 - 0.167

q = 0.833

V(X) = npq

V(X) = 72 × 0.167 × 0.833

V(X) = 10.02

V(X) [tex]\simeq[/tex] 10

∴ X [tex]\sim[/tex] N ([tex]\mu = 12, \sigma^2 =10[/tex])

⇒ [tex]\hat p = \dfrac{X}{n} \sim N ( p, \dfrac{pq}{n})[/tex]

where p = 0.167 and [tex]\dfrac{pq}{n}[/tex] = [tex]\dfrac{0.167 \times 0.833}{72}[/tex] = 0.00193

∴ P(at most 10% of rolls are 6's)

i.e

P ([tex]\hat p[/tex] ≤ 0.10)

Answer:

It´s rational

Step-by-step explanation:

27,14159 = 2714159/100000

Rational

Answer:

the answer is 4.115 x 10^5

Step-by-step explanation:

hope that helps

Answer:

2.0

Step-by-step explanation:

Answer:

B. 2.0 miles

Step-by-step explanation:

Tony ran 1/2 of a mile for 1/4 of an hour.

First, to make it easier, change each fraction into decimals:

1/2 = 0.5

1/4 = 0.25

It takes Tony 0.25 hours to run 0.5 miles.

You are solving for 1 hours worth. Multiply 4 to both terms:

0.25 hr x 4 = 1 hr

0.5 miles x 4 = 2.0 miles

B. 2.0 miles is your answer.

~

Answer and Step-by-step explanation: The Trapezoidal and Simpson's Rules are method to approximate a definite integral.

Trapezoidal Rule evaluates the area under the curve (definition of integral) by dividing the total area into trapezoids.

The formula to calculate is given by:

[tex]\int\limits^a_b {f(x)} \, dx = \frac{b-a}{2n}[f(x_{0})+2f(x_{1})+2f(x_{2})+...+2f(x_{n-1})+f(x_{n})][/tex]

The definite integral will be:

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{9-4}{2.8}[2+2.\sqrt{5} +2.\sqrt{6} +2.\sqrt{7}+2.\sqrt{8}+3][/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{5}{16}[25.3193][/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = 7.9122[/tex]

Simpson's Rule divides the area under the curve into an even interval number of subintervals, each with equal width.

The formula to calculate is:

[tex]\int\limits^a_b {f(x)} \, dx = \frac{b-a}{3n}[f(x_{0})+4f(x_{1})+2f(x_{2})+...+2f(x_{n-2})+4f(x_{n-1})+f(x_{n})][/tex]

The definite integral will be:

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{9-4}{3.8}[2+4.\sqrt{5} +2.\sqrt{6} +4.\sqrt{7} +4\sqrt{8} +3][/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{5}{24}[40.7398][/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = 8.4875[/tex]

Calculating the definite integral by using the Fundamental Theorem of Calculus:

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \int\limits^9_4 {x^{\frac{1}{2} }} \, dx[/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{2.\sqrt[]{x^{3}} }{3}[/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = \frac{2.\sqrt[]{9^{3}} }{3}-\frac{2.\sqrt[]{4^{3}} }{3}[/tex]

[tex]\int\limits^9_4 {\sqrt{x} } \, dx = 12.6667[/tex]

Comparing results, note that Simpson's Rule is closer to the exact value, i.e., gives better approximation to the exactly value calculated by the fundamental theorem.

Answer:

The answer is

Step-by-step explanation:

Equation of a line is y = mx + c

where

m is the slope

c is the y intercept

To find the equation of the parallel line we must first find the slope of the original line

From the question

y = - 4x + 3

Comparing with the general equation above

Slope / m = - 4

Since the lines are parallel their slope are also the same

Slope of parallel line = (-3 , 2) and slope

- 4 is

We have the final answer as

Hope this helps you

Answer:

y=-4x-14

Step-by-step explanation:

Answer:

add the expressions 832

Answer: y=4/5x+12

Step-by-step explanation:

Slope-intercept form: y=ax+b, where a is the slope and b is the y-intercept

When a line is parallel to another line, they will have the same slope. So the slope of the new line is also 4/5.

y=4/5x+b

To find the y-intercept, we simply plug in the point it passes through.

y=4/5x+b

8=4/5(-5)+b

8=-4+b

b=12

y=4/5x+12

Hope this helps!! :)

Please let me know if you have any question

Answer:

-3,-1,1,3,5

Step-by-step explanation:

Answer:

The answer to the problem is 5.

Answer:

Step-by-step explanation:

[tex]-3-\left(-8\right)\\\\\mathrm{Apply\:rule}\:-\left(-a\right)=a\\=-3+8\\\\\mathrm{Add/Subtract\:the\:numbers:}\:\\-3+8\\=5[/tex]

Answer:

see below

Step-by-step explanation:

a  R=k(L/A)

Substitute what we know into the equation

.08 = k (100/.05)

.08 = k 2000

Divide each side by 2000

.08/2000 = k

.00004 = k

R=.00004 (L/A)

b R=.00004 (L/A)

We know L = 4000 and A = .02

R=.00004 (4000/.02)

R = 8

Answer:

A. 5 < AC < 10

Step-by-step explanation:

If ∆ABC is a right angled triangle, we use the Pythagoras formula:

c² = a² + b²

Where c = longest side

When given sides AB, AC and BC, the formula becomes:

AB² = AC² + BC²

Where AB = Longest side

In the question,

AB = 10 and BC = 5.

10² = AC² + 5²

AC² = 10² - 5²

AC² = 100 - 25

AC² = 75

AC = √75

AC = 8.6602540378

Therefore, the expression that is always true = A. 5 < AC < 10

Answer:

First recipe needs = 3/8

Second recipe needs = 5/8

Step-by-step explanation:

Given:

Two recipes needs butter = 1 pound

One recipe needs 1/4 pound more

Find:

Each recipe needs butter

Computation:

Assume;

First recipe needs = x

Second recipe needs = x + (1/4)

Two recipes needs butter = First recipe needs + Second recipe needs

x + x + (1/4) = 1

2 x = 1 - (1/4)

2 x = 3/4

x = 3/8 pound

First recipe needs = 3/8

Second recipe needs = x + (1/4) = (3/8) + (1/4) = 5/8

Second recipe needs = 5/8

Complete question :

A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this, the company randomly chooses 6 small cities and offers the candy bar at different prices. Using candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below:

City - - - - - - - Price ($) -- - Sales

River City - - 1.30 - - - - - - 100

Hudson - - - 1.60 - - - - - 90

Ellsworth - - - 1.80 - - - - - 90

Prescott - - - - 2.00 - - - - 40

Rock Elm - - 2.40 - - 38

Stillwater - - 2.90 - - 32

Answer:

78.39%

Step-by-step explanation:

Given the data :

Price (X) :

1.30

1.60

1.80

2.00

2.40

2.90

Sales (y) :

100

90

90

40

38

32

The percentage of the total variation in candy bar sales explained by the regression model can be obtained from the value of the Coefficient of determination(R^2) of the regression model. The Coefficient of determination is a value which ranges between 0 - 1 and gives the proportion of variation in the dependent variable which can be explained by the dependent variable.

R^2 value is obtained by getting the squared value of R(correlation Coefficient).

The R value obtained using the online R value calculator on the data is : - 0.8854

Hence, R^2 = (-0.8854)^2 = 0.7839

Expressing 0.7839 as a percentage ;

0.7839 × 100 = 78.39%

Answer: D. This variable is a discrete numerical variable that is ratio-scaled.

Step-by-step explanation:

A Discrete variables are variables which are countable in a finite amount of time. For example, you can count the amount of money in your bank wallet, but same can’t be said for the money you have deposited in eveyones bank account as this is infinite.

So the number of times an individual changes job in a five years period is a perfect example of  a discrete numerical variable that is ratio scaled because it can be counted.

Answer:

Step-by-step explanation:

From the given information:

r = 10 cos( θ)

r = 5

We are to find the  the area of the region that lies inside the first curve and outside the second curve.

The first thing we need to do is to determine the intersection of the points in these two curves.

To do that :

let equate the two parameters together

So;

10 cos( θ) = 5

cos( θ) = [tex]\dfrac{1}{2}[/tex]

[tex]\theta = -\dfrac{\pi}{3}, \ \ \dfrac{\pi}{3}[/tex]

Now, the area of the  region that lies inside the first curve and outside the second curve can be determined by finding the integral . i.e

[tex]A = \dfrac{1}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} (10 \ cos \ \theta)^2 d \theta - \dfrac{1}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \ \ 5^2 d \theta[/tex]

[tex]A = \dfrac{1}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} 100 \ cos^2 \ \theta d \theta - \dfrac{25}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \ \ d \theta[/tex]

[tex]A = 50 \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \begin {pmatrix} \dfrac{cos \ 2 \theta +1}{2} \end {pmatrix} \ \ d \theta - \dfrac{25}{2} \begin {bmatrix} \theta \end {bmatrix}^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}}[/tex]

[tex]A =\dfrac{ 50}{2} \int \limits^{\dfrac{\pi}{3}}_{-\dfrac{\pi}{3}} \begin {pmatrix} {cos \ 2 \theta +1} \end {pmatrix} \ \ d \theta - \dfrac{25}{2} \begin {bmatrix} \dfrac{\pi}{3} - (- \dfrac{\pi}{3} )\end {bmatrix}[/tex]

[tex]A =25 \begin {bmatrix} \dfrac{sin2 \theta }{2} + \theta \end {bmatrix}^{\dfrac{\pi}{3}}_{\dfrac{\pi}{3}} \ \ - \dfrac{25}{2} \begin {bmatrix} \dfrac{2 \pi}{3} \end {bmatrix}[/tex]

[tex]A =25 \begin {bmatrix} \dfrac{sin (\dfrac{2 \pi}{3} )}{2}+\dfrac{\pi}{3} - \dfrac{ sin (\dfrac{-2\pi}{3}) }{2}-(-\dfrac{\pi}{3}) \end {bmatrix} - \dfrac{25 \pi}{3}[/tex]

[tex]A = 25 \begin{bmatrix} \dfrac{\dfrac{\sqrt{3}}{2} }{2} +\dfrac{\pi}{3} + \dfrac{\dfrac{\sqrt{3}}{2} }{2} + \dfrac{\pi}{3} \end {bmatrix}- \dfrac{ 25 \pi}{3}[/tex]

[tex]A = 25 \begin{bmatrix} \dfrac{\sqrt{3}}{2 } +\dfrac{2 \pi}{3} \end {bmatrix}- \dfrac{ 25 \pi}{3}[/tex]

[tex]A = \dfrac{25 \sqrt{3}}{2 } +\dfrac{25 \pi}{3}[/tex]

The diagrammatic expression showing the area of the region that lies inside the first curve and outside the second curve can be seen in the attached file below.