Find the perimeter and area of the figure pictured below. Let {eq}x {/eq} = 7 inches. Use the {eq}pi {/eq} key on your graphing calculator. Round your answer to the nearest hundredth.

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To find the perimeter and area of the figure pictured below, we need to break it down into simpler shapes and then add up their perimeters and areas.

First, we can see that the shape consists of a semicircle and a rectangle. We can start by finding the perimeter of the rectangle, which is simply the sum of its four sides:

Perimeter of rectangle = 2(length + width) = 2(9 + x) = 2(9 + 7) = 32 inches

Next, we need to find the perimeter of the semicircle. The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. Since we only have a semicircle, we need to divide the circumference by 2:

Circumference of semicircle = (1/2)(2πr) = πr

To find the radius of the semicircle, we can use the Pythagorean theorem. The height of the rectangle is x, and the diameter of the semicircle is also x, so the radius of the semicircle is half of x, or x/2. The width of the rectangle is 9 inches, so the length of the hypotenuse of the right triangle formed by the rectangle and the diameter of the semicircle is (9 + x) inches. Using the Pythagorean theorem:

(9 + x)^2 = x^2 + (2x)^2/4 81 + 18x + x^2 = 5x^2/4 324 + 72x + 4x^2 = 5x^2 x^2 - 72x - 324 = 0 (x - 18)(x + 18) = 0 x = 18 (since x can't be negative)

Therefore, the radius of the semicircle is 9 inches, and the circumference of the semicircle is:

Circumference of semicircle = πr = π(9) = 9π inches

Now we can add up the perimeters of the rectangle and semicircle to find the total perimeter:

Total perimeter = Perimeter of rectangle + Circumference of semicircle = 32 + 9π inches

To find the area of the figure, we need to find the area of the rectangle and the area of the semicircle, and then add them together:

Area of rectangle = length x width = 9(x) = 9(7) = 63 square inches Area of semicircle = (1/2)πr^2 = (1/2)π(9)^2 = 40.5 square inches Total area = Area of rectangle + Area of semicircle = 63 + 40.5 = 103.5 square inches

Therefore, the perimeter of the figure is \approx imately 59.13 inches (rounded to the nearest hundredth), and the area of the figure is \approx imately 103.5 square inches.