for the simple harmonic motion equation d=4sin(8pi), what is the maximum displacement from the equilibrium position?

The polynomial function of the least degree with integral coefficients that has the zeros [tex]1 + 3i[/tex], [tex]-2i[/tex] is f(x) = x⁴ - 2x³ + 14x² - 8x + 40.

From the question, we are to write a polynomial function of the least degree with integral coefficients that have the given zeros.

The given zeros are [tex]1+3i[/tex],[tex]-2i[/tex].

If [tex]1 + 3i[/tex] and [tex]-2i[/tex]are zeros of a polynomial function with integral coefficients, then their conjugates [tex]1 - 3i[/tex] and [tex]2i[/tex] are also zeros of the function.

To find the polynomial function, we can use the fact that if r is a zero of a polynomial function, then (x - r) is a factor of the function. Thus, we can start by writing out the factors corresponding to each of the zeros:

[tex](x - (1 + 3i))(x - (1 - 3i))(x - (-2i))(x - 2i)[/tex]

Next, we can simplify these factors by multiplying them out:

[tex][(x - 1) - 3i][(x - 1) + 3i](x + 2i)(x - 2i)\\= [(x - 1)^2 - (3i)^2](x^2 - (2i)^2)[/tex]

= [(x - 1)² + 9](x² + 4)

Expanding the terms, we get:

(x² - 2x + 10)(x² + 4)

Multiplying out the factors, we obtain:

x⁴ - 2x³ + 10x² + 4x² - 8x + 40

Simplifying this expression, we get:

x⁴ - 2x³ + 14x² - 8x + 40

Hence, the polynomial function is f(x) = x⁴ - 2x³ + 14x² - 8x + 40

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According to this inequality, the number of persons attending a theatrical inequality  play, denoted by p, must be less than or equal to 275 in order for everyone to have a seat.

In mathematics, an inequality is a non-equal connection between two expressions or values. As a result, imbalance leads to inequity. In mathematics, an inequality connects two values that are not equal. Inequality is not the same as equality. When two values are not equal, the not equal symbol is typically used (). Various disparities, no matter how little or huge, are utilised to contrast values. We resuming our current status quo. Yet a variety of things lead to inequality: Negative values on both sides are split or added.

a. The disparity in representing the voting age in the United States is as follows:

a ≥ 18

In order to be eligible to vote, a voter's age, denoted by a, must be more than or equal to 18 years old.

b. The inequality used to depict a theater's seating capacity is:

p ≤ 275

According to this inequality, the number of persons attending a theatrical play, denoted by p, must be less than or equal to 275 in order for everyone to have a seat.

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The name of this circle is Circle R.

The name of the radius is RB.

The name of the diameter is RI or RD.

In Mathematics, a circle can be defined as a closed, two-dimensional curved geometric shape with no edges or corners. Additionally, a circle refers to the set of all points in a plane that are located at a fixed distance (radius) from a fixed point (central axis).

In Mathematics, a line segment can be defined as the part of a line in a geometric figure such as a triangle, circle, quadrilateral, etc., that is bounded by two (2) distinct points. Additionally, a line segment typically has a fixed length.

In this context, we have the following names;

Circle R.

Radius = RB.

Diameter = RD or RI.

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Answer:

The total number of batteries that Emily buys can be expressed as:

8p + 30b

Here, 8p represents the total number of batteries in the packs that Emily buys, and 30b represents the total number of batteries in the boxes that Emily buys. By adding these two terms together, we can find the total number of batteries that Emily buys.

Answer:

After 15 years, the amount in the account will be $21.611.32, assuming the interest is compounded annually.

Step-by-step explanation:

To calculate the amount after 15 years, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the amount, P is the principal (initial amount), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).

In this case, P = $12,000, r = 0.04 (4% expressed as a decimal), n = 1 (compounded annually), and t = 15 years.

Plugging in the values, we get:

A = $12,000(1 + 0.04/1)^(1*15)

A = $12,000(1.04)^15

A = $12,000(1.801)

A = $21,611.32

Therefore, after 15 years, the amount in the account will be $21.611.32, assuming the interest is compounded annually.

The linear function with the given properties is: f(x) = 8 - 6x.

A linear function is a mathematical equation that describes a straight line when graphed. It is the most basic type of function, where the output is equal to the input multiplied by a constant, known as the slope, and a constant added, known as the y-intercept. Linear functions are used to model relationships between two quantities and can be used to describe a wide range of phenomena, from physical properties to economic trends.

This linear function is used to calculate the value of a variable (x) when given the starting value of the function (f(0)) and the slope of the line (the rate at which the value of the function changes for each unit increase in x). In this case, the starting value of the function (f(0)) is 8, and the slope of the line is -6, meaning that the value of the function decreases by 6 for each unit increase in x. Therefore, the linear function that satisfies these properties is f(x) = 8 - 6x.

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Answer:

no

Step-by-step explanation:

The solutions to the indicated quantities are x = 25, MIK = 25 and KIE = 50 degrees

The value of x

From the question, we have the following angle that can be used in our computation:

Angle MIE = 75

Using the above as a guide, we have the following:

x + 2x = 75

Evaluate

3x = 75

So, we have

x = 25

The angle MIK

From the figure, we have

MIK = x

So, we have

MIK = 1 * 25

MIK = 25

The angle KIE

From the figure, we have

KIE = 2x

So, we have

KIE = 2 * 25

KIE = 50

Hence, the measure of angle KIE is 50 degrees

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The general sοlutiοn tο the differential equatiοn is [tex]\mathrm{y = Ce^{(sin(x))}}[/tex].

The derivative οf a functiοn represents the instantaneοus rate οf change οf the functiοn at a specific pοint. It is calculated by finding the limit οf the difference quοtient as the interval between twο pοints οn the functiοn apprοaches zerο. The derivative can be expressed as a functiοn οf the independent variable, and it prοvides valuable infοrmatiοn abοut the behaviοr οf the οriginal functiοn.

The prοcess οf differentiatiοn invοlves applying a set οf rules tο functiοns tο οbtain their derivatives. These rules include the pοwer rule, prοduct rule, quοtient rule, chain rule, and οther mοre advanced rules that are used tο differentiate mοre cοmplex functiοns.

Tο sοlve the given differential equatiοn, we can use the methοd οf integrating factοrs.

First, we can rewrite the equatiοn as:

y'cοsx = ysinx + sin(175x)

Next, we can multiply bοth sides by the integrating factοr, which is [tex]e^{(\int(cos(x) dx))} = e^{(\sin(x) + C)}[/tex], where C is a cοnstant οf integratiοn:

[tex]\mathrm {e^{(sin(x)) }y'cosx = e^{(sin(x))} ysinx + e^{(sin(x))}sin(175x) + Ce^{(sin(x))}}[/tex]

Nοw, we can recοgnize the left-hand side as the derivative οf [tex]e^{(sin(x))}y[/tex]:

[tex](e^{(sin(x))y)}' = e^{(sin(x))} y' + cos(x) e^{(sin(x))}y[/tex]

Substituting this intο the abοve equatiοn, we get:

[tex]\mathrm{(e^{(sin(x))y)}' = e^{(sin(x)) }ysinx + e^{(sin(x))}sin(175x) + Ce^{(sin(x))}}[/tex]

[tex]cos(x) e^{(sin(x))}y = e^{(sin(x))y)}'[/tex]

Separating variables and integrating bοth sides, we get:

[tex]\int e^{sin(x) }dy/y = \int cos(x) dx[/tex]

ln|y| + C = sin(x) + C'

where C' is anοther cοnstant οf integratiοn.

Therefοre, the general sοlutiοn tο the differential equatiοn is:

[tex]\mathrm{|y| = e^{(sin(x)) }e^{(C' - sin(x))}}[/tex]

[tex]\mathrm{y = \± e^{(C' - sin(x) + sin(x))}}[/tex]

[tex]\mathrm{y = Ce^{(sin(x))}}[/tex]

where C is an arbitrary cοnstant.

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Answer:

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

Step-by-step explanation:

To find the equation of a line parallel to -4y = 32 - 6x, we first need to rewrite this equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

-4y = 32 - 6x

Divide both sides by -4:

y = -8 + (3/2)x

So the slope of this line is 3/2. Since we want a line parallel to this one, it will have the same slope of 3/2.

Now we can use the point-slope form of a linear equation to find the equation of the line that passes through (0,7) with a slope of 3/2:

y - y1 = m(x - x1)

where m = 3/2 and (x1,y1) = (0,7)

y - 7 = (3/2)(x - 0)

Simplifying:

y - 7 = (3/2)x

y = (3/2)x + 7

So the equation of the line parallel to -4y = 32 - 6x that passes through the point (0,7) is y = (3/2)x + 7.

Answer:

radius r = 22.1 m

height h = 5.91 m

slant height s = 22.8765841 m

volume V = 3022.73898 m^3

lateral surface area L = 1588.30288 m^2

base surface area B = 1534.38527 m^2

total surface area A = 3122.68815 m^2

Step-by-step explanation:

For the sample size of 20, the 95% confidence interval is (-4.87, 27.16), which is narrower than the previous interval due to the larger sample size.

In statistics, mean is a measure of central tendency that represents the average value of a set of numbers. It is calculated by summing up all the numbers in the set and dividing the sum by the total number of values in the set. Mean is also commonly referred to as the arithmetic mean. It is a commonly used statistical measure in many fields including finance, economics, social sciences, and more.

Here,

To calculate the z-scores associated with each student, we first need to calculate the sample mean and standard deviation for the Distance to Work variable:

Sample mean: (0+0+5+10+15+30+32)/7 = 10.71

Sample standard deviation: √(((0-10.71)² + (0-10.71)² + (5-10.71)² + (10-10.71)² + (15-10.71)² + (30-10.71)² + (32-10.71)²)/6) = 10.72

Now we can calculate the z-scores for each student:

Student 1: (0 - 10.71) / 10.72 = -0.94

Student 2: (0 - 10.71) / 10.72 = -0.94

Student 3: (5 - 10.71) / 10.72 = -0.53

Student 4: (10 - 10.71) / 10.72 = -0.07

Student 5: (15 - 10.71) / 10.72 = 0.40

Student 6: (30 - 10.71) / 10.72 = 1.80

Student 7: (32 - 10.71) / 10.72 = 1.98

Student 8: (8 - 10.71) / 10.72 = -0.25

Student 9: (36 - 10.71) / 10.72 = 2.37

Student 10: (9 - 10.71) / 10.72 = -0.16

Student 11: (22 - 10.71) / 10.72 = 1.05

Student 12: (10 - 10.71) / 10.72 = -0.07

Student 13: (15 - 10.71) / 10.72 = 0.40

Student 14: (28 - 10.71) / 10.72 = 1.54

Student 15: (12 - 10.71) / 10.72 = 0.12

Student 16: (6 - 10.71) / 10.72 = -0.44

Student 17: (0 - 10.71) / 10.72 = -0.94

Student 18: (10 - 10.71) / 10.72 = -0.07

Student 19: (30 - 10.71) / 10.72 = 1.80

Student 20: (0 - 10.71) / 10.72 = -0.94

To identify potential outliers, we can look for z-scores that are more than 2 standard deviations away from the mean (i.e., greater than 2 or less than -2). From the list above, we can see that Students 6, 7, 9, and 19 have z-scores greater than 2, indicating that they may be potential outliers.

For the second sample of seven data points (0, 0, 5, 10, 15, 30, 32), the mean is 11.14 and the standard deviation is 12.05. Using a t-distribution with 6 degrees of freedom (n-1), the 95% confidence interval is (-7.54, 29.82).

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The height of the tree is equal to 210 ft to the nearest foot using the sine rule.

The sine rule is a relationship between the size of an angle in a triangle and the opposing side.

angle opposite of the tree = 56 - 26 = 30

angle opposite of the shadow = 180 - 90 - 56= 34

by application of sine rule:

sin 30/h = sin 34/235

h = (235 × sin 30)/sin 34 {cross multiplication}

h = 117.5/0.5592

h = 210.1216 ft.

Therefore, the height of the tree is equal to 210 ft to the nearest foot using the sine rule.

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Answer:Not dividing the entire side of the equation.

Not distributing properly using F.O.I.L.

Confusing the rules for adding and subtracting negative numbers with the rules for multiplying and dividing negative numbers.

Multiplying exponents with the base.

Step-by-step explanation:

Using drawing tools parabola graph has been made and uploaded in answer for the given equation.

Standard form of parabola having vertical axis of symmetry given by:

[tex](y - k)^2 = 4p(x - h)[/tex]

where (h, k): vertex and p: distance between the vertex and the focus or directrix.

Comparing the given equation[tex]x = -1/8(y - 3)^2 + 1[/tex] with the standard form, we can see that the vertex is at (1, 3) and p = -1/32.

Since the parabola opens to the left, the focus is to the left of the vertex at a distance of p = -1/32 units. Thus, the focus is located at (-1/32, 3).

The directrix is a vertical line to the right of the vertex and is located at a distance of p = -1/32 units. Thus, the directrix is the vertical line x = 33/32.

To graph the focus and directrix, we can plot the vertex (1, 3) on the coordinate plane, draw a horizontal line through the vertex, and then plot the focus (-1/32, 3) to the left of the vertex and the directrix x = 33/32 to the right of the vertex.

Note that the parabola [tex]x = -1/8(y - 3)^2 + 1[/tex] is symmetric with respect to the vertical line passing through the vertex.

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In the given demand equation, the value of x and corresponding price p that maximize revenue are x = 31.8 and p = $514.60.

To find the value of x and corresponding price p that maximize revenue, we first need to determine the revenue function. Revenue (R) is given by the product of price (p) and quantity demanded (x), so:

R(x) = xp

Substituting the given demand equation for p, we get:

R(x) = x(1/2 + ² - 20x + 1200)

Simplifying and rearranging:

R(x) = 1/2x + ²x - 20x² + 1200x

R(x) = -20x² + (1200 + ²)x + 1/2x

To find the value of x that maximizes revenue, we take the derivative of R(x) with respect to x, set it equal to zero, and solve for x:

dR/dx = -40x + (1200 + ²) + 1/2 = 0

Solving for x:

x = (1200 + ² + 1/2) / 40

x = 31.8

To find the corresponding price, we substitute this value of x back into the demand equation for p:

p = 1/2 + ² - 20(31.8) + 1200

p = $514.60

Therefore, the value of x and corresponding price p that maximize revenue are x = 31.8 and p = $514.60.

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Answer:

answer is B

Step-by-step explanation:

-4x-7x=-11x

+3-8=-5

=-11x-5

answer: B -11x-5

[tex] \sf \implies \: - 4x +3 - 7x - 8[/tex]

[tex] \sf \implies \: - 11x +3 - 8[/tex]

[tex] \sf \implies \: - 11x - 5[/tex]

Hence, Option B is correct!!

Answer:

answer is 11.60

Step-by-step explanation:  

formula is C^2 = a^2 + b^2

c = 13

a = 8

b = ?

169 = 64 + x^2

135 = x^2

I hope this helps!

Answer:

  (x -7)² = 100

Step-by-step explanation:

You want to rewrite the equation x² -51 = 14x by completing the square.

When we're finished, the equation will be of the form ...

  (x +a)² = b

Expanding this, we have ...

  x² +2ax +a² = b

That is, the value of 'a' is half the coefficient of x when the equation is written with the x-terms together.

Adding 51 -14x to both sides of the given equation, we get ...

  x² -51 = 14x

  x² -14x = 51

Now, we can see that a=-14/2 = -7. We can add a² = (-7)² = 49 to both sides to complete the square:

  x² -14x +49 = 51 +49

  (x -7)² = 100 . . . . . . . rewritten equation

The conclusion is that if A is true, B is true, then C must be true as well.

What is statement ?

In logic and mathematics, a statement is a declarative sentence that is either true or false. It is an assertion about a fact or a relationship between facts. A statement can be a simple sentence, such as "The sun is shining," or a complex one, such as "If it rains tomorrow, I will stay indoors."

If the statement "If A AND B, then C" is true, and if its reverse statement is also true, then the converse of the statement is "If C, then A AND B."

If A is true and B is true, then A AND B is also true. Therefore, by the original statement, C must also be true.

Therefore, the conclusion is that if A is true, B is true, then C must be true as well.

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Answer:

Opposite (or a any synonyms)
Addition (Or sum)
division
inverse

Step-by-step explanation:

The first one is the most complicated, theres so many words can fit. Probably try with an synonyms of reverse like "Opposite"  
For the 2nd,3rd and 4th options:

Addition (Or sum) since is the reverse of subtraction
division  since is the reverse of multiplication
inverse since those functions (Exponent and radicals) are also inverse of each other

Answer:

B) 2≤x≤8.

Step-by-step explanation:

Add 6 to all sides of the inequality.

0+6≤3x-6+6≤18+6

6≤3x≤24

Divide all sides by 3.

2≤x≤8

The correct answer is B) 2≤x≤8.

Answer:

To estimate the dealership's cost for the car, we need to first calculate the dealership's selling price. We know that the car has an MSRP of $34,995.99, and we know that dealerships typically have a 20% markup on selling price.

So, to find the dealership's selling price, we can use the formula:

Selling price = MSRP + (Markup percentage * MSRP)

Substituting the values we have:

Selling price = $34,995.99 + (20% * $34,995.99)

Selling price = $34,995.99 + $6,999.20

Selling price = $41,995.19

Therefore, we estimate that the dealership paid $41,995.19 - the 20% markup - for the car.

To check this estimate, we can calculate the dealership's cost using the formula:

Cost = Selling price - Markup

Substituting the values we have:

Cost = $41,995.19 - (20% * $41,995.19)

Cost = $41,995.19 - $8,399.04

Cost = $33,596.15

This is very close to the MSRP of $34,995.99, which suggests that our estimate is reasonable.

The proof that the quadrilateral is a rectangle is shown below

From the question, we have the following parameters that can be used in our computation:

A = (-2, 3)

B = (-4, 1)

C = (2, -1)

D = (0, -3)

From the graph, we have the following lengths

AB = √8

CD = √8

BD = √32

AC = √32

The above shows that opposite sides are equal

From the graph, we have the following slopes

AB = 1

CD = 1

BD = -1

AC = -1

The above shows that opposite sides are have equal slopes and adjacent sides are their slopes to be opposite reciprocals

Hence, the quadrilateral is a rectangle

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Step-by-step explanation:

The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept.

Given the slope m = -2 and the y-intercept (a, b) = (-5, -4), we can substitute these values into the equation to get:

y = mx + b

y = -2x + (-4)

y = -2x - 4

Therefore, the equation in slope-intercept form is y = -2x - 4.

Answer:

Step-by-step explanation:

Let’s denote the production levels of coal, electricity and steel as x, y and z respectively. We can set up a system of equations to represent the inter-industry demand for each industry’s output.

For coal: 0.02x + 0.04y + 0.10z = x For electricity: 0.30x + 0.04y = y For steel: 0.30x + 0.02y + 0.04z = z

Solving this system of equations gives us x = (50/3)y and z = (25/2)y.

The new sales demand for coal is $2 billion (double the original), for electricity is $3 billion (triple the original) and for steel is $6 billion (an increase of 50%). Substituting these values into our equations gives us:

(50/3)y = $2 billion y = $3 billion (25/2)y = $6 billion

Solving these equations gives us y = $3 billion, x = $5 billion and z = $18.75 billion.

So to satisfy the new demand, the coal industry should produce at a level of $5 billion, the electric industry should produce at a level of $3 billion and the steel industry should produce at a level of $18.75 billion.

Answer:

1) S=31.68 yd  A=174.23 [tex]yd^{2}[/tex]

2) S=36.65 in A=256.56 [tex]in^{2}[/tex]

3) S=8.9 ft A=13.35 [tex]ft^{2}[/tex]

4)S=17.28 in. A=51.84 [tex]in^{2}[/tex]

5)S=25.31 yd. A=126.54 [tex]yd^{2}[/tex]

6)S=3.4ft A=5.11 [tex]ft^{2}[/tex]

Step-by-step explanation:

Notes:

360 was divided by 1/2, making the denominator 180. This is why there is a 1/2 at the front.

You also do not need to simplify if you're using a calculator.

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

- r is the radius

- θ is [tex]\frac{angle\pi }{180}[/tex]

S= r θ

-r is the radius

- θ is [tex]\frac{angle\pi }{180}[/tex]

1)

S=r θ

S=11([tex]\frac{165\pi }{180}[/tex])........................................plug in values

S=11( [tex]\frac{11\pi }{12}[/tex]).........................................simplify

S=31.68 yd....................................solve and round

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

A=[tex]\frac{1}{2}[/tex]([tex]11^{2}[/tex]) [tex]\frac{165\pi }{180}[/tex]....................................plug in values

A=  =[tex]\frac{1}{2}[/tex]([tex]121[/tex])  [tex]\frac{11\pi }{12}[/tex]....................................simplify

A=174.23 [tex]yd^{2}[/tex].....................................solve and round.

2)

S=r θ

S=14([tex]\frac{150\pi }{180}[/tex])...........................................plug in values

S=  =14(  [tex]\frac{5\pi }{6}[/tex]).........................................simplify

S=36.65 in........................................solve and round

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

A=[tex]\frac{1}{2}[/tex]([tex]14^{2}[/tex])  [tex]\frac{150\pi }{180}[/tex]......................................plug in values

A=  =[tex]\frac{1}{2}[/tex]([tex]196[/tex])  [tex]\frac{5\pi }{6}[/tex]......................................simplify

A=256.56 [tex]in^{2}[/tex]....................................solve and round.

3)

S=r θ

S=3([tex]\frac{170\pi }{180}[/tex]).........................................plug in values

S=  3(  [tex]\frac{17\pi }{18}[/tex]).......................................simplify

S=8.9 ft..........................................solve and round

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

A=[tex]\frac{1}{2}[/tex]([tex]3^{2}[/tex]) [tex]\frac{170\pi }{180}[/tex]........................................plug in values

A=  =[tex]\frac{1}{2}[/tex]([tex]9[/tex])  [tex]\frac{17\pi }{18}[/tex].......................................simplify

A=13.35 [tex]ft^{2}[/tex]........................................solve and round.

4)

S=r θ

S=6([tex]\frac{165\pi }{180}[/tex]).......................................plug in values

S=6( [tex]\frac{11\pi }{12}[/tex]).......................................simplify

S=17.28 in....................................solve and round

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

A=[tex]\frac{1}{2}[/tex]([tex]6^{2}[/tex]) [tex]\frac{165\pi }{180}[/tex].....................................plug in values

A=  =[tex]\frac{1}{2}[/tex]([tex]36[/tex])  [tex]\frac{11\pi }{12}[/tex]..................................simplify

A=51.84 [tex]in^{2}[/tex].....................................solve and round.

5)

S=r θ

S=10([tex]\frac{145\pi }{180}[/tex]).........................................plug in values

S=10(  [tex]\frac{29\pi }{36}[/tex]).........................................simplify

S=25.31 yd.......................................solve and round

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

A=[tex]\frac{1}{2}[/tex]([tex]10^{2}[/tex]) [tex]\frac{145\pi }{180}[/tex].......................................plug in values

A=  =[tex]\frac{1}{2}[/tex]([tex]100[/tex])  [tex]\frac{29\pi }{36}[/tex]....................................simplify

A=126.54 [tex]yd^{2}[/tex].....................................solve and round.

6)

S=r θ

S=3([tex]\frac{65\pi }{180}[/tex])..........................................plug in values

S=  3(  [tex]\frac{13\pi }{36}[/tex]).......................................simplify

S=3.4ft............................................solve and round

A=[tex]\frac{1}{2}[/tex]([tex]r^{2}[/tex]) θ

A=[tex]\frac{1}{2}[/tex]([tex]3^{2}[/tex]) [tex]\frac{65\pi }{180}[/tex].......................................plug in values

A=  =[tex]\frac{1}{2}[/tex]([tex]9[/tex])[tex]\frac{13\pi }{36}[/tex]......................................simplify

A=5.11 [tex]ft^{2}[/tex].........................................solve and round.