The Equation of line is 3x - y + 24 =0.
Given:
The table is given by:
X: 0 1 2 3 4 5 6 7
Y:-8 -5 -2 -1 -1 2 5 8
To find the equation of line by using the above data.
Now, According to the question:
From the table, We have
We know that,
Slope (m) = [tex]\frac{y_{2}-y_{1} }{x_{2} - x_{1} }[/tex]
and, slope (m) = [tex]\frac{-5-(-8)}{1-0}[/tex]
m = (-5 + 8)/ 1 = 3
So, [tex]y - y_{1} = m (x - x_{1} )[/tex]
y - 0 = 3 (x - (-8))
y = 3(x + 8)
y = 3x + 24
Hence, The Equation of line is 3x - y + 24 =0.
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A local high school runs a game at a fundraising event. In this game, marbles are randomly picked froma bag. The bag contains 5 red marbles, 4 blue marbles, and 1 green marble. If a green marble is drawn, you win $10. If the blue marble is drawn, you win $2, and if you draw a red marble you win nothing. The game costs $3 to play. Find the expected value of playing.
Given:
5 red marbles
4 blue marbles
1 green marble
Upon choosing green marble, we win $10
Upon choosing blue marble, we win $2
Upon choosing red marble, we win $0
It costs $3 to play each game.
To find:
The expected value of playing the game
Step by step solution:
Firstly we need to calculate the probablity of occuring of each event:
P (Choosing a red marble) = 5/10 = 1/2
P (Choosing a blue marble) = 4/10 = 2/5
P (Choosing a green marble) = 1/10 = 1/10
We will now associate the money related to each case:
= 1/2 × $0 + 2/5 × $2 + 1/10 × $10
= 0 + $ 4/5 + $1
= $ 1.8
One interior angle of a triangle is 98.5°, and the other two interior angles are congruent. What is the degree measure of one of the congruent angles?
81.5°
90°
40.75°
49.25°
Answer:
(c) 40.75°
Step-by-step explanation:
Given an isosceles triangle with one angle 98.5°, you want the measures of each of the other two angles.
Angle sum theoremThe sum of angles in a triangle is 180°, so we have ...
2x + 98.5° = 180° . . . . . . x is the unknown angle measure
x +49.25° = 90° . . . . . . divide by 2
x = 40.75° . . . . . . . . . . subtract 49.25°
The measure of one of the congruent angles is 40.75°.
Technology required. Ramps in a parking garage need to beboth steep and safe. The maximum safe incline for a ramp is8.5 degrees. Is this ramp safe? If not, provide dimensions thatwould make the ramp safe.
Ok, so:
Let me draw the situation here below:
Ok, let's find the angle first:
For this, we shall make use of the trigonometric functions:
In this case, the most useful function which could help us to solve this problem is tan(x). This function relations the opposite side of the angle and its adjacent side, like this:
tan(x) = opposite side / adjacent side
So, if we replace the values:
tan(x) = 15/95
tan(x) = 0,15789474
To find x, we could use the inverse function of tan(x). This one is called arctan(x).
So, arctan(tan(x)) = arctan(0,15789474)
And this is:
x = 8.97 degrees
Now, we can affirm that the angle is 8.97 degrees, which is bigger than 8.5 degrees.
Now, what should be the length of the bottom for the ramp to be safe?
Let me draw the situation:
We know that the ramp is safe if the maximum safe incline is 8.5°. So, what should be the value of x for this occurs?
We use the trigonometric function tan(x) again.
tan(8.5°) = 15/x
Remember that tan(8.5°) is a number
Then, x = 15/tan(8.5°), and this is 100.36
So, we conclude that the lenght of the bottom should be at least, 100.36
Find the complement of the set given that U= {3,4,5,6,7,8,9,10,11} (use the roster method to write the set. Enter empty or 0 the empty set.) {5,7,9,10}
Let
[tex]\begin{gathered} U=\mleft\lbrace3,4,5,6,7,8,9,10,11\mright\rbrace \\ \text{and} \\ A=\mleft\lbrace5,7,9,10\mright\rbrace \end{gathered}[/tex]Then,
[tex]\begin{gathered} A^C=U-A=\lbrace3,4,5,6,7,8,9,10,11\rbrace-\lbrace5,7,9,10\rbrace=\mleft\lbrace3,4,6,8,11\mright\rbrace \\ \Rightarrow A^C=\lbrace3,4,6,8,11\rbrace \end{gathered}[/tex]Thus, the answer is {3,4,6,8,11}
The cost of 5 gallons of ice cream has a variance of 64 with a mean of 34 dollars during the summer.
What is the probability that the sample mean would differ from the true mean by less than 1.1 dollars if a sample of 38 5-gallon pails is randomly selected? Round your answer to four decimal places.
The probability is [(X - μ) < 1.1] = 0.6046.
What is probability ?A probability is a numerical representation of the likelihood or chance that a specific event will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.
Given: σ² = 64
Mean μ = 34
To find: Probability[(X - μ) < 1.1]
Computation: Standard deviation σ = √σ²
Standard deviation σ = √64
Standard deviation σ = 8
Probability[(X - μ) < 1.1] = Probability[-1.1 < (X - μ) < 1.1]
Probability[(X - μ) < 1.1] = Probability[-1.1/(8/√38) < (X - μ) < 1.1/(8/√38)]
Using z table
The probability = [(X - μ) < 1.1] = 0.6046.
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M&Ms Data Set 20 in Appendix B lists data from 100 M&Ms, and 8% of them are
brown. Use a 0.05 significance level to test the claim of the Mars candy company that the
percentage of brown M&Ms is equal to 13%.
The rejection region, or area of the rejection zone, is the significance level. The test's tail determines the direction of the rejection region. When the significance threshold is unknown, it is typically considered to be 5%.
What is rejection region ?A collection of values for the test statistic for which the null hypothesis is rejected constitutes a critical zone, sometimes referred to as the rejection region. For example, if the observed test statistic is in the critical range, the null hypothesis is rejected and the alternative hypothesis is accepted.
Consider the crucial value as a cut-off point at a particular importance level. If a test statistic falls on one side of the crucial value and leads to the null hypothesis being accepted, a test statistic falling on the other side will lead to the null hypothesis being rejected.
A set of values for the test statistic for which the null hypothesis is accepted is referred to as a confidence interval and is also referred to as the acceptance area. For example, if the observed test statistic is within the confidence interval, the null hypothesis is accepted and the alternative hypothesis is rejected.
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what is the effect on the graph of the function f(x) = x² when f(x) is changed to f(x) + 9A) shifted up B) shifted left C) shifted rightD) shifted down
When we add a constant on a f(x) graph we have
• f(x) + a ⇒ graph shifted up
,• f(x) - a ⇒ graph shifted down
Therefore, let's see if it's shifting up or down, we have
[tex]f(x)+9[/tex]9 is positive, then it's shifting up 9 units.
What is the value of y?
2
4
6
8
Answer: Ima say 6
Explanation: I don't have an explanation
Is the coordinate (5,-3) a solution to the following system of equations?
y=2x-13
2x+6y=-8
Yes, the coordinate is a solution to both equations
No, the coordinate is not a solution for either equaiton
No, the coordinate is only a solution for y-2x-13
ordinate is only a solution for 2x+6y=-8
Option (a) is correct.
What is a equation?statement of equality between two expressions consisting of variables and/or numbers. In essence, equations are questions, and the development of mathematics has been driven by attempts to find answers to those questions in a systematic way.
Given that,
The given equation are
y=2x-13 ........(1)
2x+6y=-8 ........(2)
Solve the equation by elimination method
From equation (1)
y-2x = -13
From equation (2)
Now,
y-2x = -13
6y+2x = -8
adding both equation
7y = -21
y = -3
substitute the value of y in equation (1)
y = 2x-13
-3 = 2x-13
-3+13 = 2x
2x = 10
x = 5
Hence, The coordinate (5,-3) is a solution to both equations.
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Find the constant of proportions k as a fraction in simplest form. The enter an equation for the relation ships between x and y.The constant of proportions, k is equal to?The equation is y = ?
Notice that for every 6 units increment in the x-values in the table, there is a 2 units increment in the y-values.
Therefore, the constant of proportions is:
[tex]\begin{gathered} \frac{2}{6}\rightarrow\frac{1}{3} \\ k=\frac{1}{3} \end{gathered}[/tex]To find the equation, we'll use this constant as the slope. Using this, point
(6, 2) of the table and the slope-point form:
[tex]\begin{gathered} y-2=\frac{1}{3}(x-6) \\ \rightarrow y-2=\frac{1}{3}x-2 \\ \rightarrow y=\frac{1}{3}x \end{gathered}[/tex]This way the equation is:
[tex]y=\frac{1}{3}x[/tex]Find the value of x and y in which each quadrilateral mist be a parallelogram.
Recall the property of parallelograms that tell us that the diagonals intercept producing equal segments in each of them. Then, as the diagonals intercept in the midpoint of each diagonal, then we can create the following equations and solve them one at a time (making the measure of one segment equal the value on the other half of the diagonal):
5 y - 4 = 36 then we solve here for y
add 4 to both sides
5 y = 36 + 4
5 y = 40
divide both sides by 5
y = 40 / 5
y = 8
Now the second equation:
4 x - 10 = 18
add 10 on both sides
4 x = 28
divide both sides by 4 to isolate x
x = 28 / 4
x = 7
8. For every 1,000 feet above sea level, the air temperature decreases by 4°F.The highest natural point in Texas is Guadalupe Peak at 8,751 feet. If thetemperature at sea level is 76°F, what is the temperature near the summit at8,000 feet? (Example 4)
Answer
The temperature at an altitude of 8,000 feet is 44° F
SOLUTION
Problem Statement
The question tells us that air temperature drops by 4 degrees Fahrenheit for every 1,000 feet above sea level. We are asked to calculate the temperature at 8,000 feet, close to the summit of Guadalupe Peak.
Solution
To solve this question, we simply compute the temperature changes for each 1,000 increase in height and then generalize. With the generalization, we can find the temperature at any height.
The temperature at 0 feet is 76°F (At sea level)
The temperature at 1,000 feet is 76°F - 4°F = 72° F
The temperature at 2,000 feet (2 x 1,000 feet) is 76°F - 4°F - 4°F (76°F - 2 x 4°F)
The temperature at 3,000 feet (3 x 1,000 feet) is 76°F - 4° F - 4° F - 4° F (76° F - 3 x 4° F)
And so on.
We can see a pattern developing and with this pattern, we can make a generalization about the temperature decrease for every increase in altitude.
[tex]\begin{gathered} The\text{ temperature at }n\times1,000\text{ feet is (}76^o-n\times4^o)F \\ \text{where,} \\ n=A\text{ whole number starting from zero} \end{gathered}[/tex]Thus, we can solve the question by substituting n = 8 for an altitude of 8,000 feet. This is done below:
8 x 1000 feet = 8,000 feet is:
[tex]\begin{gathered} 76^oF-8\times4^{o\text{ }}F \\ 76^oF-32^oF \\ =44^oF \end{gathered}[/tex]Final Answer
The temperature at an altitude of 8,000 feet is 44° F
Two buses leave Philadelphia at the same time and travel in opposite directions. The north bound bus travels
at a rate of 55 mph. The southbound bus travels at a rate of 45 mph. When will they be 400 miles apart?
Answer:
4 hours
Step-by-step explanation:
55x + 45x = 400
100x = 400
x = 400/100
x = 4
Really need help solving this Struggling It’s from my trig prep book
Start making the graph of the situation
from this, we can understand that x is Coreys' initial distance, z is Coreys' final distance, and y will be how many feet had Corey to step back in order to gain a better view.
Using the red triangle we find x through the tan of the given angle
[tex]\begin{gathered} \tan \theta=\frac{op}{ad} \\ \tan 68=\frac{80}{x} \\ x=\frac{80}{\tan 68} \\ x\approx32.32 \end{gathered}[/tex]Using the blue triangle we find z through the tan of the given angle the same way as before
[tex]\begin{gathered} \tan \theta=\frac{op}{ad} \\ \tan 41=\frac{80}{z} \\ z=\frac{80}{\tan 41} \\ z\approx92.03 \end{gathered}[/tex]finally, find y as the difference between z and x
[tex]\begin{gathered} z=x+y \\ y=z-x \\ y=92.03-32.32 \\ y=59.71 \end{gathered}[/tex]Corey had to go back 59.71 ft to gain a better view.
Write this in a equation. 1. three increased by 6 times a number2. ten more than twice a number3. the difference between 4 times a number and 14. 12 less than 3 times a number is negative 65. three times the difference of a number and 126. 2 more than 5 times a number is 17
Answer:
3. the difference between 4 times a number and 1
[tex]4x-1[/tex]5. three times the difference of a number and 12
[tex]3(x-12)[/tex]Explanation:
Writing the equation of the given expressions;
Let x represent the missing number;
3. the difference between 4 times a number and 1
[tex]4x-1[/tex]5. three times the difference of a number and 12
[tex]3(x-12)[/tex]A runner sprinted 365.21 ft to finish a race.Use the table of facts to find the distance she sprinted in yards.Round your answer to the nearest tenth.
we have the following:
[tex]365.21ft\cdot\frac{1y}{3ft}=121.74[/tex]Therefore, the answer is 121.7 yards
Hallie has 10 times as many pages to read for her
homework assignment as Janet. Altogether, they have
to read 264 pages. How many pages
does each girl
have to read?
Answer:
24
Step-by-step explanation:
10x + x = 264
11x = 264
x = 264 ÷ 11
x = 24
therefore hallie has 240 pages and Janet 24
total 264
the area of the parallelogram below is 59.9m2,What is the height
The height of the parallelogram is 6.97m.
How to calculate the height?It is important to kite that the aea of a parallelogram is calculated as:
= Base × Height
In this case, the base is given as 8.6m. Therefore, the height will be:
Area = Base × Height
59.9 = 8.6 × Height
Now, we will find the height
Height = 59.9 / 8.6
Height = 6.97m
The complete question is written below.
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The area of the parallelogram is 59.9m² and.the vase is 8.6m,What is the height
The students in Hugh Logan's math class took the Scholastic Aptitude Test. Their math scores areshown below. Find the mean score. Round your answer to the nearest tenth.563 524 357 347 637358 351 528 470 482pls be fast
Answer:
[tex]461.7\Rightarrow\text{ Option \lparen D\rparen}[/tex]Explanation: We have to find the mean of the total scores shown, the formula for the mean is as follows:
[tex]A=\frac{a_1+a_2+a_3+a_4+...+a_N}{N}\rightarrow(1)[/tex]Using the formula (1) the mean of the score is calculated as follows:
[tex]\begin{gathered} A=\frac{563+524+357+347+637+358+351+528+470+482}{10} \\ \\ A=\frac{4617}{10}=441.7 \\ \\ A=461.7\rightarrow\text{ \lparen D\rparen} \end{gathered}[/tex]Therefore the answer is Option(D).
14. If the surface area of the cone below is 628.32 m2, find its volume.
You have a cone with surface area of 628.32 m². In order to find the volume of the cone you use the following formula:
V= 1/3 π r² h
r: radius of the base of the cone = 15m/2 = 7.5m
h: heigth of the cone
Then, you have to calculate first the height of the cone. To do that, you use the information about the surface area. You use the following formula:
S = πr² + πrl
l: diagonal of the cone
Then, you have to find l, and then you can calculate hYou solve for l from the surface area formula:
l = (S - πr²)/πr = (628.32 - π(7.5)²)/(π(7.5))
l = 19.16
To calculate the heigth of the cone you use the Pythagoras theorem, just as follow:
h = √(l²-r²) = √((19.16)²-(7.5)²) = 17.63m
Next, you can replace the values of h and r into the formula for the Volume:
V= 1/3 π r² h = (1/3)π (7.5m)²(17.63m) = 1038.5m³
Hence, the volume of the given cone is 1038.49 m3
In this diagram, ABAC – AEDF. If thearea of ABAC = 6 in2, what is thearea of AEDF?DE2 inFB3 inСArea = [?] in?Enter a decimal rounded to the tenths.
Answer:
8/3 square inches
Explanation:
First, we need to get the height of the triangle BAC
Area of triangle = 1/2 * base * height
6 = 1/2 * 3 * height
12 = 3 * height
Height = 12/3
The height of triangle BAC is 4inches
Next is to get the height of EDF. Since they are both similar hence;
2/3 = h/4
3h = 2 * 4
3h = 8
h = 8/3
h = 8/3in
Hence the height of triangle EDF is 8/3in
Get the area of triangle EDF
Area of triangle EDF = 1/2 * 2 * 8/3
Area of triangle EDF = 8/3 square inches
Hence the area of triangle EDF is 8/3square inches
During a family trip, you share the driving with your dad. At most, you are allowed to drive for three hours. While driving, your maximum speed is 55 miles per hour. a) Write a system of inequalities describing the possible numbers of hours, t, and the distance, d, you may have to drive.
Here, we want to writw an inequality to describe the information given.
When we say at most, it means the expected values may be less than or equal to but can never be more than
Mathematically;
[tex]\text{distance d = sp}eed\text{ }\times\text{ time t}[/tex]From the question, we are told that the maximum driving time is 3 hours; the inequality here will be;
[tex]t\text{ }\leq\text{ 3 hours}[/tex]For the distance d, we have;
[tex]\begin{gathered} d\text{ }\leq\text{ 3 }\times\text{ 55} \\ \\ d\text{ }\leq\text{ 165 miles} \end{gathered}[/tex]I need help with this question please. I have options available
Option B is the graph of the given function.
please help me identify what was wrong in solving it I know the right answer is 10 but i don't know what was done wrong.
Given:
The coordinates are given as,
[tex]\begin{gathered} (x_1,y_1)=(-1,2) \\ (x_2,y_2)=(-7,-6) \end{gathered}[/tex]The objective is to find the distance between the two points.
Explanation:
The general formula to find the distance between two coordinates is,
[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}\text{ . . . . . (1)}[/tex]Substitute the given coordinates in equation (1),
[tex]d=\sqrt[]{(-7-(-1))^2+(-6-2)^2}[/tex]On further solving the above equation,
[tex]\begin{gathered} d=\sqrt[]{(-6)^2+(-8)^2} \\ =\sqrt[]{36+64} \\ =\sqrt[]{100} \\ =10 \end{gathered}[/tex]Hence, the distance between the two points is 10.
Shaun estimated that the attendance at a college baseball game was 1,000. The actual Q3: attendance was 1,788. What is the percent error of Shaun's estimate? Round to the nearest whole percent.
The percentage error of Shaun's estimate is 44%
Here, we want to get the percentage error
We can get this by using the formula below;
[tex]\frac{error}{\text{actual value}}\text{ }\times\text{ 100\%}[/tex]We have the error as the difference between the estimated value and the actual value
The difference is;
[tex]1,788\text{ - 1,000 = 788}[/tex]The percentage error is;
[tex]\frac{788}{1,788}\text{ }\times\text{ 100 \% = 44.07 }[/tex]This is approximately 44%
Complete the work to solve for y7 (22y + 5) - 13 = 2 4 - 1 + 101/4 + 2 - 1/2 = 1 4 - 1 + you
Answer:
11/2
Explanation:
Given:
To find:
The value of y
So we have;
[tex]\frac{11}{5}=\frac{2}{5}y[/tex]Let's go ahead and apply the Division property of equality by dividing both sides of the equation by 2/5;
[tex]\begin{gathered} \frac{\frac{11}{5}}{\frac{2}{5}}=\frac{\frac{2}{5}}{\frac{2}{5}}y \\ \frac{11}{5}\div\frac{2}{5}=y \\ \frac{11}{5}*\frac{5}{2}=y \\ \frac{11}{2}=y \\ \therefore y=\frac{11}{2} \end{gathered}[/tex]So the value of y is 11/2
M is the midpoint of line AB. Endpoint A is (4, 2). Midpoint M is (6, 0). Endpoint B = ?
The B points are ( 8, -2 ) of Endpoint B when M is the midpoint of line AB.
What is midpoint of a segment?
The midway of a line segment in geometry is where it meets the other end. The centroid of the segment and the endpoints, it is equally spaced from both endpoints. It cuts the section in half.M is the midpoint of line AB.
Endpoint A is (4, 2).
Midpoint M is (6, 0)
Let mid points are ( x,y) and A points are ( x₂ , y₂ ) and B points are ( x₁ , y₁)
x = x₁ + x₂/2 , y = y₁ + y₂/2
6 = x₁ + 4/2 , 0 = y₁ + 2/2
12 = x₁ + 4 , 0 = y₁ + 2
x₁ = 12 - 4 , y₁ = -2
x₁ = 8 , y₁ = -2
So, the B points are ( 8, -2 )
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The number of Americans over age 100 can be approximated by P(t) = 0.07e^0.54t, where is P(t) measured in millions and t is measured in decades. Also, t=0 corresponds to 2000.
a. How many Americans over age 100 were there in 2000?
b. How fast was the number of Americans over age 100 changing in 2000? Use the correct units.
There are 7 % of Americans over age 100 who were there in 2000 and the number of Americans growing by 54% over age 100 changed in 2000.
What is an exponential function?An exponential function is defined as a function whose value is a constant raised to the power of an argument is called an exponential function.
It is a relation of the form y = aˣ in mathematics, where x is the independent variable
The number of Americans over age 100 can be approximated by
[tex]P(t) = 0.07e^{0.54t}[/tex]
where is P(t) measured in millions and t is measured in decades.
Also, t = 0 corresponds to 2000.
⇒ [tex]P(t) = 0.07e^{0.54t}[/tex]
Substitute the t = 0 in the above function, and we get
⇒ [tex]P(t) = 0.07e^{0.54\times0}[/tex]
⇒ P(t) = 0.07 e°
⇒ P(t) = 0.07
Convert decimals into fractions, and we get
⇒ P(t) = 7 / 100
This represents 7 out of 100 which is 7 % of Americans over age 100 who were there in 2000.
Thus, the number of Americans growing by 54% over age 100 changed in 2000.
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part 2..number 2...evaluate the expressions using numbers and evaluate the expression for the numbers provided...please read the direction to make sure I'm understanding what's asked of me..I'm a 7th grader can u please help me with my math summer package and explain it in a way that a 7th grader can understand it
Given:
Each story is 13ft tall.
Explanation:
a) To find: The height of the skyscraper if it has 55 stories.
Since the number of stories increases the height of the skyscraper also increases.
That is,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=55\times13 \\ =715ft \end{gathered}[/tex]b) To find: The height of the skyscraper if it has 65 stories.
The height of the skyscraper would be,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=65\times13 \\ =845ft \end{gathered}[/tex]c) To find: The height of the skyscraper if it has 75 stories.
The height of the skyscraper would be,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=75\times13 \\ =975ft \end{gathered}[/tex]d) To find: The equation of the height if it has f stories
The height of the skyscraper would be,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=f\times13 \\ h=13f \end{gathered}[/tex]Final answer:
a) The height of the skyscraper if it has 55 stories is 715ft.
b) The height of the skyscraper if it has 65 stories is 845ft.
c) The height of the skyscraper if it has 75 stories is 975ft.
d) The expression that represents the height of f stories is h = 13f.
What is the area shaded sector to the nearest 10th of the square centimeter?
The area of a sector of a circumference is given by the following formula:
[tex]A=\frac{\theta r^2}{2}[/tex]Where r is the radius of the circumference, and θ is the angle of the sector in radians.
The radius is already known: 15cm.
We need to estimate the angle of the shaded region. The shaded region and the sector whose angle is 72° form together a 180° angle. (This is according to the figure. That is not said in an explicit way but we will need to assume that since there is not enough information to calculate the area if it is otherwise.)
Then, the angle of the shaded area, plus 72° is 180°:
[tex]\begin{gathered} \theta+72^o=180^o \\ \\ \theta=180^o-72^o \\ \\ \theta=108^o \end{gathered}[/tex]Now, before applying the formula we need to express the angle in radians. Recalling that 180° is equal to π radians:
[tex]\theta=108^o\cdot\frac{\pi\text{ rad}}{180^o}=\frac{108}{180}\pi\text{ rad}[/tex]Now, we have the angle in radians. We can use the equation:
[tex]\begin{gathered} A=\frac{\theta r^2}{2} \\ \\ A=\frac{108}{180}\cdot\frac{\pi\cdot(15\operatorname{cm})^2}{2} \\ \\ A\approx212.1\text{ cm}^2 \end{gathered}[/tex]