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 Circle The radius of the circle is r=3.6 feet. Now, we can use the formula for the area of a circle.

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.

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

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.

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.