Exercise 51 Page 847

Let's analyze the given shaded region.

We are asked to find its area. Notice that it is equal to the difference between the area of the equilateral triangle and the circle. c Area of Region = c Area of Triangle - c Area of CircleThe radius of the circle is r=3.6 feet. Now, we can use the formula for the area of a circle.

A_\text{circle}=\pi r^2

▼

Substitute 3.6 for r and evaluate

Substitute

r= 3.6

A_\text{circle}=\pi ({\color{#009600}{3.6}})^2

CalcPowProd

Calculate power and product

A_\text{circle}=12.96\pi

π ≈ 3.14

A_\text{circle}\approx 12.96({\color{#0000FF}{3.14}})

Multiply

A_\text{circle}\approx 40.6944

RoundDec

Round to 2 decimal place(s)

A_\text{circle}\approx 40.69

Therefore, the area of the circle is about 40.69 square feet. Now we will find the area of the triangle. Let's analyze an equilateral triangle with three heights and the inscribed circle.

Notice that the altitudes of the equilateral triangle are also its medians. From the Centroid Theorem, the centroid C is two-thirds of the distance from each vertex to the midpoint of the opposite side. This tells us that BC is two times larger than AC. BC=2* AC=2* 3.6=7.2 ft We can find the height h=AB.

AB=AC+BC

▼

Substitute values and evaluate

SubstituteII

AC= 3.6, BC= 7.2

AB= 3.6+ 7.2

AddTerms

Add terms

AB=10.8

Since the triangle is equilateral, each of its angle measures 60^(∘). Now, let's use the trigonometric ratios in △ BAD.

sin(m∠ ADB)=AB/BD

SubstituteII

m∠ ADB= 60^(∘), AB= 10.8

sin 60^(∘)=10.8/BD

▼

Solve for BD

sin 60^(∘) ≈ 0.866

0.866≈10.8/BD

MultEqn

LHS * BD=RHS* BD

0.866 BD ≈ 10.8

DivEqn

.LHS /0.866.=.RHS /0.866.

BD ≈ 10.8/0.866

CalcQuot

Calculate quotient & Use a calculator

BD ≈ 12.4711

RoundDec

Round to 2 decimal place(s)

BD≈ 12.47

Therefore, the side length of the equilateral triangle is about s=12.47 feet. Let's use the formula for the area of an equilateral triangle.

A_\text{triangle}=\dfrac{s^2\sqrt{3}}{4}

▼

Substitute 12.47 for s and evaluate

Substitute

s= 12.47

A_\text{triangle} = \dfrac{(12.47)^2\sqrt{3}}{4}

CalcPow

Calculate power

A_\text{triangle} = \dfrac{155.5009\sqrt{3}}{4}

UseCalc

Use a calculator

A_\text{triangle} = 67.33386\ldots

RoundDec

Round to 2 decimal place(s)

A_\text{triangle} \approx 67.33

The area of the equilateral triangle is about 67.33 square feet. Finally, let's find the area of the shaded region. c Area of Region = c Area of Triangle - c Area of Circle ⇓ c Area of Region = 67.33- 40.69=26.64 This tells us that the area of the shaded region is about 26.64 square feet.

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