Question

manuel is planning to apply stucco to a building whose dimensions are shown below. the front of the building has a window measuring 64 in by 63 in and a door measuring 3 ft by 7 ft. the left and right sides of the building each have a window measuring 20 ft by 3 ft. the back of the building has no window or door. use the given information to answer the questions. in the figure below, each tab shows a different view of the building. front right view back left view conversion facts for length 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 yard (yd) = 36 inches (in) (a) assuming the roof, windows, and door require no stucco, what is the area in square feet that needs stucco? (do not round any intermediate computations and give your answer as a whole number.) \\(\square\text{ ft}^2\\) (b) manuel will use stucco that is sold in bags. each bag

Explanation:

Step1: Calculate total surface area of the building (excluding roof, windows, door)

First, find the surface area of the front, back, left, and right sides.
- Front and back: The front has a window, back has no window/door.
- Front dimensions: 18 ft (height) × 30 ft (width)? Wait, no, looking at the diagram, front right view: height 18 ft, and the building's length (depth) is 30 ft, and the side (left/right) has a window of 20 ft × 3 ft. Wait, let's re-express:
- Front: area = 18 ft × 30 ft = 540 sq ft. But front has a window: 64 in × 63 in. Convert inches to feet: 64 in = \( \frac{64}{12} \) ft, 63 in = \( \frac{63}{12} \) ft. Area of front window: \( \frac{64}{12} \times \frac{63}{12} = \frac{64 \times 63}{144} = \frac{4032}{144} = 28 \) sq ft. So front area to stucco: 540 - 28 = 512 sq ft.
- Back: 18 ft × 30 ft = 540 sq ft (no window/door), so back area: 540 sq ft.
- Left and right sides: Each left/right side: height 18 ft, width? Wait, the diagram shows 8 ft? Wait, no, the front right view: the side (yellow roof) and the building's side: maybe the left and right sides have dimensions 18 ft (height) × 8 ft (width)? Wait, no, the problem says "left and right sides of the building each have a window measuring 20 ft by 3 ft". Wait, maybe the left and right sides: area of each side (excluding window) is (18 ft × 8 ft) - (20 ft × 3 ft)? Wait, no, 20 ft by 3 ft: but 20 ft is longer than 18 ft? That can't be. Wait, maybe the window on left/right is 3 ft by 7 ft? Wait, the problem says: "the left and right sides of the building each have a window measuring 20 ft by 3 ft". Wait, maybe I misread. Wait, the problem statement: "the front of the building has a window measuring 64 in by 63 in, a door measuring 3 ft by 7 ft. The left and right sides of the building each have a window measuring 20 ft by 3 ft. The back of the building has no window or door."
Ah, door is 3 ft × 7 ft. So front has window (64in×63in) and door (3ft×7ft). Wait, I missed the door! Front has window and door. So front area: 18 ft (height) × 30 ft (width) = 540 sq ft. Door area: 3 ft × 7 ft = 21 sq ft. Window area: 28 sq ft (as before). So front area to stucco: 540 - 28 - 21 = 540 - 49 = 491 sq ft? Wait, no, maybe the front's height is 18 ft, and the width (horizontal) is 30 ft? Wait, the diagram shows front right view: height 18 ft, and the building's length (depth) is 30 ft, and the side (left/right) has a window of 20 ft × 3 ft. Wait, let's start over:

Total surface area (excluding roof) is front + back + left + right.

- Front:
- Height (h) = 18 ft, Width (w) = 30 ft (assuming the building's width is 30 ft, depth is 30 ft? Wait, the diagram has 18 ft (vertical), 30 ft (horizontal, length), and 8 ft? No, the front right view: the yellow part is roof, and the building's front face: height 18 ft, and the side (left/right) has a window of 20 ft × 3 ft. Wait, maybe the building's dimensions: front/back height 18 ft, width (front/back) 30 ft, and the left/right sides (depth) 8 ft? No, the problem says "left and right sides of the building each have a window measuring 20 ft by 3 ft". So left and right sides: each has area (18 ft × 30 ft)? No, that doesn't make sense. Wait, perhaps the building is a rectangular prism with:
- Height: 18 ft
- Length (front to back): 30 ft
- Width (left to right): 8 ft? Wait, the diagram shows "front right view" with a horizontal arrow of 8 ft, and vertical 18 ft, and depth 30 ft. So:
- Front and back faces: height 18 ft, width 30 ft (area = 18×30 = 540 sq ft each)
- Left and right faces: height 18 ft, width 8 ft (area = 18×8 = 144 sq ft each)
Now, windows/door:
- Front face: window (64in×63in) and door (3ft×7ft)
- Window: 64 in = 64/12 ft = 16/3 ft ≈ 5.333 ft; 63 in = 63/12 ft = 21/4 ft = 5.25 ft. Area: (16/3)×(21/4) = (16×21)/(12) = (336)/12 = 28 sq ft.
- Door: 3 ft × 7 ft = 21 sq ft. So front area to stucco: 540 - 28 - 21 = 491 sq ft.
- Back face: no window/door, so area = 18×30 = 540 sq ft.
- Left and right faces: each has a window of 20 ft × 3 ft. Each left/right face area: 18×8 = 144 sq ft. Window area: 20×3 = 60 sq ft. So each left/right area to stucco: 144 - 60 = 84 sq ft. Two sides: 84×2 = 168 sq ft.

Now total area to stucco: front + back + left + right = 491 + 540 + 168 = 1199? Wait, that can't be. Wait, maybe I messed up the dimensions. Let's re-express the building's dimensions:

From the diagram: front right view has 18 ft (vertical), 30 ft (horizontal, length), and 8 ft (another dimension). Wait, maybe the building is a rectangular prism with length (L) = 30 ft, width (W) = 8 ft, height (H) = 18 ft. Then:

- Front and back faces: area = H × L = 18×30 = 540 sq ft each.
- Left and right faces: area = H × W = 18×8 = 144 sq ft each.

Now, front face has window (64in×63in) and door (3ft×7ft). Back face has no window/door. Left and right faces each have window (20ft×3ft).

So:

- Front: 540 - (window area) - (door area)
- Back: 540
- Left: 144 - (window area)
- Right: 144 - (window area)

Window on front: 64in = 64/12 ft = 16/3 ft, 63in = 63/12 ft = 21/4 ft. Area: (16/3)(21/4) = 28 sq ft (as before).

Door: 3×7=21 sq ft.

Windows on left/right: 20×3=60 sq ft each.

So:

Front: 540 - 28 - 21 = 491

Back: 540

Left: 144 - 60 = 84

Right: 144 - 60 = 84

Total area: 491 + 540 + 84 + 84 = 491 + 540 = 1031; 84 + 84 = 168; 1031 + 168 = 1199 sq ft? Wait, but the problem says "assuming the roof, windows, and door require no stucco (do not round any intermediate computations and give your answer as a whole number)". Wait, maybe I made a mistake in dimensions. Wait, the left/right window is 20 ft by 3 ft, but the left/right face height is 18 ft, so 20 ft is taller than 18 ft? That can't be. So maybe the left/right window is 3 ft by 7 ft? No, the problem says left/right windows are 20 ft by 3 ft. Wait, maybe the building's height is 20 ft? No, the diagram says 18 ft. Wait, perhaps the front/back height is 18 ft, and the left/right height is 20 ft? No, the diagram shows 18 ft. This is confusing. Wait, let's check the conversion and the window sizes again.

Wait, the front window is 64 in by 63 in: 64 in = 5 ft 4 in, 63 in = 5 ft 3 in. Area: 5.333 ft × 5.25 ft = 28 sq ft (correct). Door: 3×7=21 sq ft. Left/right windows: 20×3=60 sq ft each. Front area: 18×30=540. So front: 540 -28 -21=491. Back: 18×30=540. Left: 18×8 -60=144-60=84. Right: same as left:84. Total: 491+540+84+84=1199. But maybe the building's width (left/right) is 30 ft, and length is 8 ft? Let's try that:

- Front/back: H×W=18×30=540 each.
- Left/right: H×L=18×8=144 each.
Same as before. So total area is 1199? But that seems low. Wait, maybe the roof is not included, but the problem says "assuming the roof, windows, and door require no stucco" – wait, no: "assuming the roof, windows, and door require no stucco" means we exclude roof, windows, and door from stucco. So we need to calculate the exterior surface area excluding roof, windows, and door.

Wait, maybe the building is a rectangular prism with length L=30 ft, width W=8 ft, height H=18 ft. Then:

- Total exterior surface area (excluding roof) = 2×(H×L + H×W) = 2×(18×30 + 18×8) = 2×(540 + 144) = 2×684 = 1368 sq ft.

Now subtract the areas of windows and door:

- Front window: 28 sq ft
- Front door: 21 sq ft
- Two left/right windows: 2×(20×3)=120 sq ft

So total area to stucco: 1368 -28 -21 -120 = 1368 - 169 = 1199 sq ft. Yes, that matches. So the area is 1199 square feet.

Step2: Wait, but let's verify again:

Total surface area (excluding roof) = 2*(height*length + height*width) = 2*(18*30 + 18*8) = 2*(540 + 144) = 2*684 = 1368.

Subtract:

- Front window: 28
- Front door: 21
- Left window: 60
- Right window: 60

Total to subtract: 28 +21 +60 +60=169.

1368 -169=1199. So the area is 1199 square feet.

Answer:

1199