Question

the dimensions of a proposed deck are written on the floor plan. what is the area of the proposed deck? round to the nearest square foot.
○ 317 square feet
○ 413 square feet
○ 437 square feet
○ 513 square feet

Explanation:

Answer:

To find the area of the deck, we break it into three parts: a semicircle, a rectangle, and a trapezoid (or triangle and rectangle, but let's use the given dimensions).

1. Semicircle: The diameter is \( 16 + 8 - 8 = 16 \)? Wait, no. Wait, the top semicircle has a diameter of \( 16 + 8 - 8 \)? Wait, looking at the diagram: the top horizontal segment is 8 ft? Wait, no, the diagram shows: the vertical side on the right is 18 ft, the left vertical part (above the triangle) is 12 ft, so the height of the triangle is \( 18 - 12 = 6 \) ft. The base of the triangle is 8 ft, the base of the rectangle (middle part) is 16 ft, and the top semicircle: wait, the horizontal line at the top of the rectangle (the 12 ft tall part) has a length? Wait, maybe the deck is composed of:

• A semicircle with diameter \( 16 + 8 - 8 \)? No, let's re-examine. The diagram:

• The right side is 18 ft tall. The left side has a vertical segment of 12 ft, then a slant to the bottom left, with a horizontal segment of 8 ft (from the slant's bottom to the rectangle's bottom left) and 16 ft (from the rectangle's bottom left to the right bottom).

• The top part is a semicircle. The diameter of the semicircle: looking at the horizontal line at the top of the 12 ft tall rectangle: the length there is \( 16 + 8 - 8 \)? Wait, maybe the semicircle has a diameter equal to the length of the top of the middle rectangle. Wait, the middle rectangle (the 12 ft tall part) has a width of 16 ft? No, the bottom of the deck has a horizontal length of \( 16 + 8 = 24 \) ft? Wait, no, the bottom right is at 16 ft from the left (the vertical right side), and the bottom left is 8 ft to the left of that, so total bottom length is \( 16 + 8 = 24 \) ft. The middle rectangle (the 12 ft tall part) has a width of 16 ft (from the right vertical side to the left vertical side of the rectangle). The top semicircle: the diameter is 16 ft? Wait, the horizontal line at the top of the 12 ft rectangle is 16 ft? Wait, the diagram shows a dot at the left of the semicircle, a dot in the middle (8 ft from the left), and a dot at the right (16 ft from the left? No, the 8 ft is labeled between the middle dot and the right dot. Wait, the top horizontal segment (the base of the semicircle) has length \( 8 + 8 = 16 \) ft? So the diameter of the semicircle is 16 ft, so radius \( r = 8 \) ft.

• Area of semicircle: \( \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (8)^2 = 32\pi \approx 100.53 \) sq ft.

• Rectangle (middle part): Height = 12 ft, width = 16 ft. Area = \( 12 \times 16 = 192 \) sq ft.

• Trapezoid (bottom part): The bottom part is a trapezoid with bases \( 16 \) ft (right base) and \( 16 + 8 = 24 \) ft (left base? Wait, no. Wait, the bottom left is 8 ft to the left of the rectangle's bottom left, so the bottom has two parts: the rectangle's bottom (16 ft) and the triangle's base (8 ft). Wait, actually, the bottom part is a trapezoid with the two parallel sides being \( 16 \) ft (the right side, height 18 - 12 = 6 ft) and \( 16 + 8 = 24 \) ft (the left side, height 6 ft). Wait, no, the height of the trapezoid is \( 18 - 12 = 6 \) ft (the vertical distance between the bottom of the 12 ft rectangle and the bottom of the deck). The two parallel sides (bases) of the trapezoid are: the right base is 16 ft (the width of the rectangle), and the left base is \( 16 + 8 = 24 \) ft (since there's an 8 ft extension to the left). Wait, no, the trapezoid area formula is \( \frac{1}{2}(b_1 + b_2)h \). Here, \( b_1 = 16 \) ft, \( b_2 = 16 + 8 = 24 \) ft, \( h = 18 - 12 = 6 \) ft. So area = \( \frac{1}{2}(16 + 24) \times 6 = \frac{1}{2}(40) \times 6 = 120 \) sq ft.

Now, total area = semicircle + rectangle + trapezoid = \( 100.53 + 192 + 120 = 412.53 \approx 413 \) square feet.

So the answer is 413 square feet.