A sheep rancher plans to fence a rectangular pasture next to an irrigation canal. No fence will be needed along the canal, but the other three sides must be fenced. The pasture must have an area of 180,000 m² to provide enough grass for the sheep. Find the dimensions of the pasture which require the least amount of fence.

Answer:

h = 30 cm

Step-by-step explanation:

Given:

V (volume) = 69,120π cm^3

d (diameter) = 96 cm (r (radius) = 0,5 × 96 = 48 cm

Find: h (height) - ?

[tex]v = \frac{1}{3} \times \pi {r}^{2} \times h[/tex]

[tex] \frac{1}{3} \times \pi \times {48}^{2} \times h = 69120\pi[/tex]

Multiply the whole equation by 3 to eliminate the fraction:

[tex]2304\pi \times h = 69120\pi[/tex]

[tex]h = 30[/tex]

Using an equation, if Nancy worked alone, the number of hours it would take her to complete the birthday cake is 1 hour 30 minutes.

An equation is a mathematical statement that proves the equality or equivalence of two or more mathematical expressions.

Equations use the equal symbol (=) unlike algebraic expressions, which combine variables with numbers, constants, and values using mathematical operands.

The number of hours for Vanessa and Nancy working together to make a birthday cake = 2 hours

The number of hours it takes Vanessa to complete the cake working alone = x

The number of hours it takes Nancy to complete the cake alone = 3x

3x + x = 2

4x = 2

x = 0.5 = 30 minutes

The total time for Nancy to complete the cake = 3x = 1.5 (3 x 0.5)

= 1 hour 30 minutes

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Based on the given information, Yasmina should use Sally's Saddles if she expects to give 200 rides and wants to make the most profit.

To determine which company Yasmina should use to offer horseback rides at her school's Spring Fair, we need to compare the costs and revenues associated with each option.

First, let's consider Polly's Ponies. They charge $100 for a small pony and Yasmina can charge children $2 for a ride. To break even with this option, Yasmina would need to give 50 rides ($100 / $2 per ride). If she expects to give 200 rides, she would earn $400 in revenue ($2 per ride x 200 rides) and have a profit of $300 ($400 revenue - $100 rental fee).

Next, let's consider Sally's Saddles. They charge $240 for a larger horse and Yasmina can charge children $3 for a ride. To break even with this option, Yasmina would need to give 80 rides ($240 / $3 per ride). If she expects to give 200 rides, she would earn $600 in revenue ($3 per ride x 200 rides) and have a profit of $360 ($600 revenue - $240 rental fee).

Therefore, if Yasmina wants to make the same amount of profit regardless of which company she uses, she would need children to take a total of 125 rides ((($240 rental fee for Sally's Saddles - $100 rental fee for Polly's Ponies) / ($3 per ride - $2 per ride)). If she expects to give 200 rides, she should use Sally's Saddles since she will make a higher profit of $360 compared to $300 with Polly's Ponies.

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The function h(x) can be represented by -3x-9 .

An equation can be represented by a linear function. The standard form for the linear equation is: y= mx+b , for example, y=7x+6. Where:

m= the slope.

b= the constant term that represents the y-intercept.

For the given example: m=7 and b=6.

The question gives two linear equations that represent two functions: f(x)=-2x+7 and g(x)=5x-16.

For solving this you should sum both equations. See below

h(x)=f(x)+g(x)

h(x)=-2x+7 +5x-16

h(x)=-3x-9

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For the given function y = 126 \sqrt(x), dy is 3.57 if x = 9 and Δx = dx = 0.17.

To find the change in y (or dy), we need to use the formula:

dy = f'(x) * dx

where f'(x) is the derivative of y with respect to x.

The given function is y = 126 \sqrt(x).

To find the derivative, we can use the power rule and chain rule of differentiation.

y = 126x^{1/2}

Taking the derivative of y with respect to x:

dy/dx = 1/2 * 126x^(-1/2)

Simplifying, we get:

dy/dx = 63/(\sqrt(x))

Now, we can substitute x = 9 into this expression to get the value of the derivative at that point:

dy/dx = 63/(\sqrt(9)) = 63/3 = 21

Next, we can use the given value of dx, which is 0.17, to find the change in y or dy.

dy = f'(x) * dx ≈ 21 * 0.17 ≈ 3.57

Therefore, the change in y or dy is approximately 3.57.

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The required height for which the base is 6 inches and the area of 42 square inches is 7 inches.

Each piece will be in a shape of a parallelogram so we know that the area of the parallelogram is given as

area= height * base

Now we are provided with the base of the parallelogram that is 6 inches. The area of the piece is also given as 42 sq inches. So, we can solve and get the height from the above equation.

Height (h) = area/ base

Height (h)= 42/6

Height (h)= 7 inches.

Therefore, Maya should use a height of 7 inches for each piece in order to achieve a base of 6 inches and an area of 42 sq inches.

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To find the moment of inertia about the y-axis of the first-quadrant area bounded by the curve y=9−x^2 and the coordinate axes, we can use the formula:

I = ∫y² dA

where I is the moment of inertia, y is the distance from the y-axis to the infinitesimal element of area dA, and the integral is taken over the first-quadrant area.

To set up the integral, we need to express y in terms of x for the curve y=9−x². Solving for y, we get:

y = 9 - x²

The area element dA is given by:

dA = y dx

Substituting y in terms of x, we get:

dA = (9 - x²) dx

Now we can express the moment of inertia as an integral:

I = ∫y² dA
 = ∫(9 - x²)² dx     (limits of integration: x = 0 to x = 3)

To evaluate the integral, we can expand the integrand using the binomial theorem:

I = ∫(81 - 36x² + x⁴) dx
 = 81x - 12x³ + (1/5)x⁵     (limits of integration: x = 0 to x = 3)

Finally, we can substitute the limits of integration and simplify:

I = (81(3) - 12(3)³ + (1/5)(3)⁵) - 0
 = 243 - 108 + 27
 = 162

Therefore, the moment of inertia about the y-axis is 162 units^4.
To find the moment of inertia (Iy) about the y-axis for the first-quadrant area bounded by the curve y = 9 - x^2 and the coordinate axes, we need to integrate the expression for the moment of inertia using the limits of the region.

The curve intersects the x-axis when y = 0, so:

0 = 9 - x²
x² = 9
x = ±3

Since we're in the first quadrant, we're interested in x = 3.

The moment of inertia about the y-axis is given by the expression Iy = ∫x²dA, where dA is the area element. In this case, we'll use a vertical strip with thickness dx and height y = 9 - x². Therefore, dA = y dx.

Now, let's integrate Iy:

Iy = ∫x²(9 - x²) dx from 0 to 3

To solve this integral, you may need to use polynomial expansion and integration techniques:

Iy = ∫(9x² - x⁴) dx from 0 to 3
Iy = [3x³/3 - x⁵/5] from 0 to 3
Iy = (3(3)³/3 - (3)⁵/5) - (0)
Iy = (81 - 243/5)
Iy = (405 - 243)/5
Iy = 162/5

So the moment of inertia about the y-axis for the first-quadrant area bounded by the curve y = 9 - x^2 and the coordinate axes is Iy = 162/5.

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The volume of the cone is 117.3 (Round to the nearest tenth).

To find the volume of this cone with a height of 7 inches and a radius of 4 inches, and round to the nearest tenth, follow these steps:
1. Use the formula for the volume of a cone: V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height.
2. Plug in the given values: V = (1/3)π(4²)(7)
3. Calculate the volume: V = (1/3)π(16)(7) = (1/3)(112π)
4. Multiply and round to the nearest tenth: V ≈ 117.3 cubic inches

So, the volume of this cone is approximately 117.3 cubic inches.

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[tex]\sf Let \ d_1 \ and \ d_2 \ be \ the \ lengths \ of \ the \ sides \ of \ diagonals.[/tex]

[tex]\sf Given \ that \ d_1=8 \ cm[/tex]

[tex]\sf And \ d_2=10 \ cm[/tex]

[tex]\therefore\sf Area \ of \ rhombus=\dfrac{1}{2} (d_1)(d_2)=\dfrac{1}{2}(8)(10)=40 \ cm^2[/tex]

[tex]\rightarrow\boxed{\sf Area \ of \ rhombus=40 \ cm^2}[/tex]

The equation is 86.25 + 3x = 99.

Let's assume the cost for each hour Troy's car was parked in the garage is x dollars.

Since Troy spent a total of $99, we can set up an equation based on the given information.

The cost of the ticket to the basketball game is $86.25, and Troy also had to pay for 3 hours of parking. Therefore, the equation can be written as:

86.25 + 3x = 99

In this equation, 86.25 represents the cost of the ticket, 3x represents the cost of parking for 3 hours at a rate of x dollars per hour, and 99 represents the total amount Troy spent.

Therefore, the equation is 86.25 + 3x = 99.

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Answer:

16

Step-by-step explanation:

4y = 2.6

y = 0.65

20y + 3

= 20 × 0.65 + 3

= 13 + 3

= 16

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Based on the information provided, Madeline and Ben invested $1500 for their son Peter's education on the day he was born at an interest rate of 6.7% compounded quarterly. Since Peter is now 19 years old and wants to go to college, we can calculate the current value of his education fund.

To do this, we can use the formula for compound interest:

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

Where:
A = the final amount
P = the principal (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time in years

In this case, we have:

P = $1500
r = 6.7% = 0.067 (as a decimal)
n = 4 (since the interest is compounded quarterly)
t = 19 (since Peter is now 19 years old)

So, the current value of Peter's education fund is:

A = $1500(1 + 0.067/4)^(4*19)
A = $1500(1.01675)^76
A = $1500(2.4826)
A = $3,723.90

Therefore, the current value of Peter's education fund is $3,723.90. This should help Madeline and Ben determine how much more they need to save for Peter's college expenses.

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Answer:

40.7, my answer needs to be 20+ characters sooo....

The answer is A. 1. We can use L'Hopital's rule to evaluate the limit:

limit x→9√(f(x))-3/√(x)-3 = limit x→9 (f(x)-9)/(x-9) / (√(f(x))-3)/(√(x)-3)

Now, we know that f(9) = 9 and f'(9) = 4, so we can use the definition of the derivative to write:

f(x) - f(9) = f'(9)(x-9) + o(x-9)

where o(x-9) represents a term that goes to 0 faster than x-9 as x approaches 9. Plugging this into the numerator, we get:

f(x) - 9 = 4(x-9) + o(x-9)

Plugging this into the denominator, we get:

√(f(x)) - 3 = √(4(x-9) + o(x-9)) = 2√(x-9) + o(1)

√(x) - 3 = √(x-9) + o(1)

Therefore, the limit becomes:

limit x→9 (4(x-9) + o(x-9))/(√(x-9) + o(1)) / (2√(x-9) + o(1))/(√(x-9) + o(1))

Simplifying this expression, we get:

limit x→9 2(4(x-9) + o(x-9))/(√(x-9) + o(1))^2

limit x→9 8 + 2o(1)/(x-9)

As x approaches 9, the o(1) term goes to 0, so the limit becomes:

8 + 2*0/0 = 8

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1. The dollar decrease of Jane's investment account is $2,166.

2. The amount in Jane's account at the end was $5,434.

To get the decrease in dollars, we need to calculate 28.5% of $7,600.

The dollar decrease will be:

= 0.285 x $7,600

= $2,166

To find the amount, we must subtract the decrease from the initial amount. So, the end balance will be:

= $7,600 - $2,166

= $5,434

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When the diameter of the balloon is 50 cm, the radius of the balloon is increasing at a rate of approximately 0.0254 cm/s

We are given that the volume of a spherical balloon is increasing at a rate of 100 cm³/s. We need to find how fast the radius of the balloon is increasing when the diameter is 50 cm.

Let's first find the expression for the volume of the balloon in terms of its radius.

V = 4/3 πr³

Differentiating with respect to time (t), we get:

dV/dt = 4πr² (dr/dt)

We are given that dV/dt = 100 cm³/s. When the diameter of the balloon is 50 cm, the radius is 25 cm.

Substituting these values, we get:

100 = 4π(25)² (dr/dt)

Simplifying, we get:

dr/dt = 100 / (4π(25)²)

dr/dt ≈ 0.0254 cm/s

Therefore, when the diameter of the balloon is 50 cm, the radius of the balloon is increasing at a rate of approximately 0.0254 cm/s

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Answer:

True (I think)

Step-by-step explanation:

Same pattern.

A -> B -> C.

D -> E -> F.

Would be false if either one didn't share the same pattern.

Mr. Mensah's total earnings at the end of the fifteenth year of service is €1920.00 + €2520.00 = €4440.00.

To calculate Mr. Mensah's salary in the fifteenth year of service, we need to determine the pattern of salary increase over the years.

We know that Mr. Mensah's salary starts at €6400.00 and increases by €240.00 every year for the first eight years. After that, he is promoted to a new post with an annual salary of €9500.00, which increases by €360.00 every year.

Let's break it down:

For the first eight years, the salary increases by €240.00 per year:

After 1 year: €6400.00 + €240.00 = €6640.00

After 2 years: €6640.00 + €240.00 = €6880.00

...

After 8 years: €6400.00 + (8 * €240.00) = €6400.00 + €1920.00 = €8320.00

From the ninth year onwards, the salary increases by €360.00 per year:

After 9 years: €9500.00 + €360.00 = €9860.00

After 10 years: €9860.00 + €360.00 = €10220.00

...

After 15 years: €9500.00 + (7 * €360.00) = €9500.00 + €2520.00 = €12020.00

Therefore, Mr. Mensah's salary in the fifteenth year of service is €12,020.00.

To calculate Mr. Mensah's total earnings at the end of the fifteenth year of service, we need to sum up his salaries from year 1 to year 15.

For the first eight years, the total earnings can be calculated as follows:

Total earnings = (Salary in year 1 + Salary in year 2 + ... + Salary in year 8) = 8 * €240.00 = €1920.00

From the ninth year onwards, the total earnings can be calculated as follows:

Total earnings = (Salary in year 9 + Salary in year 10 + ... + Salary in year 15) = 7 * €360.00 = €2520.00

Therefore, Mr. Mensah's total earnings at the end of the fifteenth year of service is €1920.00 + €2520.00 = €4440.00.

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To find the unique function f(x) satisfying the given conditions, we will use the method of undetermined coefficients.

Assume that f(x) is a polynomial of degree n. Then, f"(x) is a polynomial of degree n-2. Therefore, x^2 f(x) is a polynomial of degree n+2.

Let's first find the second derivative of f(x):

f''(x) = (d^2/dx^2) f(x)

Since we assumed that f(x) is a polynomial of degree n, we can write:

f''(x) = n(n-1) a_n x^(n-2)

where a_n is the leading coefficient of f(x).

Now, let's substitute the given values of f(1) and f(2):

f(1) = a_n

f(2) = a_n 2^n

Therefore, we have two equations:

n(n-1) a_n = x^2 f(x)

a_n = 4

a_n 2^n = 1

Solving for n and a_n, we get:

n = 3/2

a_n = 4/3^(3/2)

Thus, the unique function f(x) that satisfies the given conditions is:

f(x) = (4/3^(3/2)) x^(3/2) - (4/3^(3/2)) x^2 + 1/2
It seems that your question is incomplete or contains some errors. However, based on the information provided, I understand that you are looking for a function f(x) that satisfies given conditions involving its second derivative and specific values of f(1) and f(2).

To assist you properly, please provide the complete and correct version of the question with all the necessary conditions.

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The area of the shaded region of the circle is A = 57,918.5 feet²

Given data ,

First, we need to find the area of the quarter circle:

Area of quarter circle = (1/4)  π ( r )²

= (1/4)  π ( 290 )²

= 66,018.5 feet²

Next, we need to find the area of the square:

Area of square = side²

= 90²

= 8,100 feet²

Now, we can find the area of the shaded section by subtracting the area of the square from the area of the quarter circle:

Area of shaded section = Area of quarter circle - Area of square

= 66,018.5 feet² - 8,100 feet²

= 57,918.5 feet²

Hence , the area of the shaded section of the field is 57,918.5 feet²

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Thus, the volume of cone for the given slant height and radius is found as:  314 cu. cm.

The distance from a cone's apex to its outer rim is referred to as the segment's slant height. It is corresponding to the hypotenuse's length of a right triangle that creates the cone.

Given data:

slant height of cone  l =  13 cm

radius r =  5 cm

Let h be the height

So, using Pythagorean theorem, find height.

l² = h² + r²

h² = l² - r²

h²= 13² - 5²

h² = 169 - 25

h = 12 cm

volume of a cone = 1/3 *π*r²*h

volume of a cone = 1/3 *3.14*5²*12

volume of a cone = 3.14*25*4

volume of a cone = 314 cu. cm

Thus, the volume of cone for the given slant height and radius is found as:  314 cu. cm.

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Answer:

Step-by-step explanation:

The expression +3(5-3) can be simplified using the order of operations (PEMDAS) as follows: first, we need to perform the exponentiation operation, which gives us 125.

Then, we need to perform the operation inside the parentheses, which gives us 6. Finally, we multiply 3 by 6, giving us 18. Therefore, the simplified expression is 125 + 18 = 143.

To further explain this solution, we use the order of operations, which is a set of rules that dictate the order in which operations must be performed when evaluating an expression. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction, which represents the order of operations from left to right.

In this expression, we first need to evaluate the exponentiation operation, which is 5 to the third power. This gives us 125. Next, we need to perform the operation inside the parentheses, which is 5-3. This gives us 2. We then multiply 3 by 2, which gives us 6. Finally, we add 125 and 6, giving us 131.

It is important to follow the order of operations when simplifying an expression to ensure that we obtain the correct result. By using PEMDAS, we can systematically simplify an expression step-by-step, avoiding any potential errors and obtaining a clear and concise solution.

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The random variable in the first situation is the number of hits for the players of a baseball team and in the second situation is the distance traveled by the tee shots in a golf game.

1) The random variable in this distribution is the number of hits for the players of a baseball team. This is a discrete random variable because hits are counted as whole numbers and cannot take on non-integer values.

2) The random variable in this distribution is the distance traveled by the tee shots in a golf game. This is a continuous random variable because the distances traveled can take on any value within a certain range, including non-integer values. The exact distance traveled by a tee shot can be measured to any degree of precision, and there are infinitely many possible distances within the range of possible outcomes. Therefore, it is a continuous random variable.

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The probability that the next contestant will win a prize is 2/5.

What is probability?

Probability is a way of calculating how likely something is to happen. It is difficult to provide a complete prediction for many events. Using it, we can only forecast the probability, or likelihood, of an event occurring. The probability might be between 0 and 1, where 0 denotes an impossibility and 1 denotes a certainty.

Experimental probability is based on the result of an experiment that has been carried out multiples times.

probability that the next contestant will win a prize = number of contestants that won in the last games / total number of contestants in the last games

=> 8/20

To transform to the simplest form. divide both the numerator and the denominator by 4

=> 2/5

Hence the probability that the next contestant will win a prize is 2/5.

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The recursive formula for this sequence is f(n) = f(n-3) + 6n - 18.

We can use the method of finite differences to find a possible recursive formula for this sequence.

First, let's compute the first few differences:

f(4) - f(1) = 6

f(7) - f(4) = 6

Since the second differences are zero, we can assume that the sequence is a quadratic sequence. Let's write it in the form f(n) = an^2 + bn + c. We can solve for the coefficients using the given values:

f(1) = a(1)^2 + b(1) + c = 6

f(4) = a(4)^2 + b(4) + c = 12

f(7) = a(7)^2 + b(7) + c = 18

Solving for a, b, and c, we get:

a = 1

b = 5

c = 0

Therefore, the recursive formula for this sequence is f(n) = f(n-3) + 6n - 18.

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The area inside the square and outside the circle is 0.86 square units.

To find the area inside the square and outside the circle, we need to first find the area of the square and the area of the circle.

Let's assume that the square has sides of length 2 units, which means its area is:

Area of square = side^2 = 2^2 = 4 square units

Now, let's assume that the circle has a radius of 1 unit, which means its area is:

Area of circle = pi * radius^2 = 3.14 * 1^2 = 3.14 square units

To find the area inside the square and outside the circle, we need to subtract the area of the circle from the area of the square:

Area inside square and outside circle = Area of square - Area of circle

                                     = 4 - 3.14

                                     = 0.86 square units

Therefore, the area inside the square and outside the circle is 0.86 square units.

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Jonana cut 24 pieces of 1/4 foot length from the 6-foot board

Let "x" be the number of pieces that Jonana cut.

Each piece is 1/4 foot long.

So, the total length of all the pieces is x * 1/4 = x/4 feet.

But the total length is also 6 feet.

So we can set up the equation:

x/4 = 6

Solving for x:

x = 24

Therefore, Jonana cut 24 pieces of 1/4 foot length from the 6-foot board.

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To determine the values of A and B, we can use the following system of equations: 1. 5A = -15 2. -2A + B = 10 This system of equations can be used to find the values of A and B.

To review the equation used in writing a partial fraction decomposition, we start with a fraction that has a denominator that can be factored into linear or quadratic factors. The partial fraction decomposition separates the fraction into a sum of simpler fractions, each with a single linear or quadratic factor in the denominator.

The equation you provided for partial fraction decomposition is:

StartFraction negative 15 x + 10 Over (5 x minus 2) squared EndFraction = StartFraction A Over 5 x minus 2 EndFraction + StartFraction B Over (5 x minus 2) squared EndFraction

This equation shows that the original fraction can be decomposed into two simpler fractions, one with a linear factor of (5x - 2) in the denominator (A/(5x - 2)), and one with a quadratic factor of (5x - 2)² in the denominator (B/(5x - 2)²).

To determine the values of A and B, we need to solve for them using a system of equations. In this case, we can use the coefficients of x in the numerator of each fraction to create the following system of equations:

5A = -15
-2A + B = 10

We get the first equation by setting the numerator of the first fraction (A/(5x - 2)) equal to -15x + 10, and the second equation by setting the numerator of the second fraction (B/(5x - 2)²) equal to -15x + 10 and subtracting the first equation from it.

Solving this system of equations gives us:

A = -3
B = 5

Therefore, the partial fraction decomposition of the original fraction is:

StartFraction negative 15 x + 10 Over (5 x minus 2) squared EndFraction = StartFraction -3 Over 5 x minus 2 EndFraction + StartFraction 5 Over (5 x minus 2) squared EndFraction

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Answer:

30 cm²

Step-by-step explanation:

Formula : [tex]Area = \frac{hb}{2}[/tex]