an outdoor garage is being built on a farm to house vehicles and equipment. the front has two congruent garage door entrances and a round window at the top.
a) the front wall will be covered in sheet metal. determine the amount of sheet metal needed, to the nearest square foot.
b) suppose the sheet metal is priced by the square metre. how many square metres will be needed for this project?
1 foot = 0.305m

Question

Accepted Answer

### Part (a)
# Explanation:
## Step 1: Calculate area of the rectangular part (excluding doors)
The total height of the rectangle is 18 ft, width is 28 ft. But we have two congruent doors. Each door has height 12 ft and width 10 ft (since they are congruent, and the first door is 10 ft wide). Wait, actually, the area of the rectangle (the lower part) is $ \text{length} \times \text{height} = 28 \times 18 $. Then subtract the area of the two doors. Each door: $ 10 \times 12 $, so two doors: $ 2 \times 10 \times 12 $.
$$
\text{Area of rectangle (lower)} = 28 \times 18 = 504
$$
$$
\text{Area of two doors} = 2 \times 10 \times 12 = 240
$$
$$
\text{Area of lower part (rectangle - doors)} = 504 - 240 = 264
$$

## Step 2: Calculate area of the triangular part
The base of the triangle is 28 ft, and we need the height of the triangle. The angle is $ 18^\circ $, and the horizontal line (the base of the triangle's right triangle) – wait, actually, the triangle at the top: the height $ h $ can be found? Wait, no, maybe the triangle is an isosceles triangle with base 28 ft, and the angle at the top is $ 18^\circ $? Wait, the diagram shows a dashed line (the base of the triangle) with length 28 ft, and the angle at the left is $ 18^\circ $. Wait, maybe the triangle is a triangle with base 28 ft, and the height $ h $ can be calculated? Wait, no, maybe the triangle is formed by the two slant sides and the top. Wait, alternatively, maybe the triangle's height is calculated using trigonometry. Wait, the angle is $ 18^\circ $, and the adjacent side? Wait, maybe the triangle is a triangle where the base is 28 ft, and the height $ h $ is such that $ \tan(18^\circ) = \frac{h}{14} $ (since the base is 28, so half is 14). Wait, $ h = 14 \times \tan(18^\circ) $. Let's calculate $ \tan(18^\circ) \approx 0.3249 $, so $ h \approx 14 \times 0.3249 \approx 4.5486 $ ft. Then area of the triangle is $ \frac{1}{2} \times 28 \times 4.5486 \approx 63.68 $ sq ft.

## Step 3: Calculate area of the circular window (subtract it)
The window is a circle with radius 2 ft, so area is $ \pi r^2 = \pi \times 2^2 = 4\pi \approx 12.57 $ sq ft.

## Step 4: Total area of sheet metal
Total area = area of lower part (rectangle - doors) + area of triangle - area of window.
$$
\text{Total area} = 264 + 63.68 - 12.57 \approx 264 + 63.68 = 327.68; 327.68 - 12.57 \approx 315.11
$$
Wait, wait, maybe I made a mistake in the triangle. Wait, maybe the triangle is a triangle with base 28 ft, and the height is calculated as follows: the angle is $ 18^\circ $, and the horizontal segment (the base of the right triangle) is 14 ft (half of 28). So $ \tan(18^\circ) = \frac{h}{14} $, so $ h = 14 \tan(18^\circ) \approx 14 \times 0.3249 \approx 4.5486 $ ft. So area of triangle: $ \frac{1}{2} \times 28 \times 4.5486 \approx 63.68 $ sq ft.

Wait, but let's recheck the lower part: the total height of the rectangle is 18 ft, width 28 ft: area 28*18=504. Then the doors: each door is 10 ft wide, 12 ft tall. So two doors: 2*10*12=240. So 504-240=264. Then the triangle: area 63.68. Then the window: area $ \pi r^2 = 4\pi \approx 12.57 $. So total area: 264 + 63.68 - 12.57 ≈ 315.11, which rounds to 315? Wait, maybe my triangle height is wrong. Wait, maybe the triangle is a different shape. Wait, the diagram shows a dashed line (the base of the triangle) with length 28 ft, and the angle at the left is 18 degrees. So maybe the triangle is a triangle where the height is calculated as follows: the horizontal line (the base of the triangle) is 28 ft, and the angle between the slant side and the horizontal is 18 degrees. So the height $ h $ of the triangle is $ \tan(18^\circ) \times $ (half of 28)? Wait, no, if the angle is at the vertex, then the triangle is isosceles with vertex angle 18 degrees, base 28 ft. Then the height $ h $ can be found by $ \tan(9^\circ) = \frac{14}{h} $? Wait, no, in an isosceles triangle, the vertex angle is 18 degrees, so the base angles are $ (180 - 18)/2 = 81^\circ $. Then, if we split the triangle into two right triangles, each with base 14 ft, angle at the base is 81 degrees, so $ \tan(81^\circ) = \frac{h}{14} $, so $ h = 14 \times \tan(81^\circ) \approx 14 \times 6.3138 \approx 88.39 $? That can't be right, because the total height of the building is 18 ft. Wait, I must have misinterpreted the diagram.

Wait, the diagram shows the total height of the rectangular part as 18 ft, and the triangular part on top. Wait, maybe the triangle is a small triangle? Wait, no, the diagram has a dashed line (the top of the rectangle) with length 28 ft, and the triangle above it. The height of the triangle: maybe the diagram's 18 degrees is the angle between the slant side and the horizontal (the dashed line). So the height $ h $ of the triangle is $ \tan(18^\circ) \times $ (the length of the horizontal segment? Wait, no, the horizontal segment is 28 ft. Wait, maybe the triangle is a triangle with base 28 ft, and height $ h $, and the angle between the slant side and the base is 18 degrees. So $ \tan(18^\circ) = \frac{h}{28} $? No, that would make $ h = 28 \times \tan(18^\circ) \approx 28 \times 0.3249 \approx 9.097 $ ft. But the total height of the building is 18 ft, so the rectangular part is 18 ft, and the triangle is on top? That can't be, because 18 + 9 would be 27 ft, but the diagram shows the rectangular part as 18 ft. Wait, I think I made a mistake. Let's re-express:

Wait, the front wall consists of:

1. A rectangle with height 18 ft, width 28 ft.

2. A triangle on top of the rectangle? No, the diagram shows the rectangle with height 18 ft, and the triangle above it? Wait, no, the diagram has the vertical side as 18 ft, and the triangle at the top. Wait, maybe the total height from the ground to the peak is 18 ft + h, but no, the vertical side is 18 ft. Wait, maybe the triangle is part of the front wall, so the front wall is a combination of a rectangle (height 18 ft, width 28 ft) minus the doors, plus a triangle (base 28 ft, height h) minus the circular window.

Wait, maybe the triangle's height is calculated as follows: the angle is 18 degrees, and the horizontal line (the top of the rectangle) is 28 ft. So the triangle is a triangle with base 28 ft, and the height h is such that $ \tan(18^\circ) = \frac{h}{x} $, but maybe x is 14 ft (half of 28). So $ h = 14 \times \tan(18^\circ) \approx 14 \times 0.3249 \approx 4.5486 $ ft. So area of triangle: $ \frac{1}{2} \times 28 \times 4.5486 \approx 63.68 $ sq ft.

Then the area of the rectangle (including the space where the doors are) is $ 28 \times 18 = 504 $ sq ft. Then subtract the area of the two doors: each door is 10 ft wide, 12 ft tall, so two doors: $ 2 \times 10 \times 12 = 240 $ sq ft. So the area of the rectangular part (excluding doors) is $ 504 - 240 = 264 $ sq ft.

Then the area of the triangular part: 63.68 sq ft.

Then subtract the area of the circular window: radius 2 ft, so area $ \pi r^2 = \pi \times 2^2 = 4\pi \approx 12.57 $ sq ft.

So total area: $ 264 + 63.68 - 12.57 \approx 315.11 $, which rounds to 315 sq ft? Wait, but maybe the triangle is not on top of the rectangle, but the front wall is a rectangle with height 18 ft, and a triangle above it, but the total height is 18 ft? No, that doesn't make sense. Wait, maybe the diagram is a front view where the lower part is a rectangle (height 18 ft, width 28 ft), and the upper part is a triangle with base 28 ft, and the height of the triangle is calculated from the angle. Wait, the angle is 18 degrees, and the horizontal line (the top of the rectangle) is 28 ft, so the triangle's height is $ \tan(18^\circ) \times $ (the length of the horizontal segment? No, maybe the triangle is a gable, so the height of the gable (triangle) is $ h $, and the base is 28 ft. Then the area of the gable is $ \frac{1}{2} \times 28 \times h $. To find $ h $, we can use the angle: the angle between the slant side and the horizontal is 18 degrees, so $ \tan(18^\circ) = \frac{h}{14} $ (since the base is 28, half is 14). So $ h = 14 \times \tan(18^\circ) \approx 14 \times 0.3249 \approx 4.5486 $ ft. So area of gable: $ \frac{1}{2} \times 28 \times 4.5486 \approx 63.68 $ sq ft.

Then the total area of the front wall (before subtracting doors and window) is the area of the rectangle (18 ft height, 28 ft width) plus the area of the gable (triangle). Wait, that makes more sense! Because the front wall is the lower rectangle (18 ft tall, 28 ft wide) plus the upper triangle (gable) with base 28 ft and height ~4.5486 ft. Then we subtract the area of the two doors and the circular window.

So:

$$
\text{Area of rectangle (lower)} = 28 \times 18 = 504
$$
$$
\text{Area of gable (triangle)} = \frac{1}{2} \times 28 \times 4.5486 \approx 63.68
$$
$$
\text{Total area (rectangle + triangle)} = 504 + 63.68 = 567.68
$$

Now subtract the area of the two doors: each door is 10 ft wide, 12 ft tall, so two doors: $ 2 \times 10 \times 12 = 240 $

Subtract the area of the circular window: $ \pi \times 2^2 = 4\pi \approx 12.57 $

$$
\text{Total sheet metal area} = 567.68 - 240 - 12.57 \approx 315.11 \approx 315 \text{ sq ft}
$$

Wait, but let's check again. Maybe the height of the triangle is different. Wait, the angle is 18 degrees, and the horizontal line (the top of the rectangle) is 28 ft. So the slant side makes 18 degrees with the horizontal. So the height of the triangle is $ \tan(18^\circ) \times 28 $? No, that would be if the base is 28, but the triangle is a right triangle. Wait, no, the triangle is isosceles, so splitting into two right triangles, each with base 14 ft, angle 18 degrees at the base. So $ \tan(18^\circ) = \frac{h}{14} $, so $ h = 14 \times \tan(18^\circ) \approx 4.5486 $ ft, which is what we had. So that's correct.

### Part (b)
# Explanation:
## Step 1: Convert square feet to square meters
We know that 1 foot = 0.305 m, so 1 square foot = $ (0.305)^2 $ square meters.

First, find the area in square feet from part (a): approximately 315.11 sq ft.

$$
\text{Area in sq meters} = 315.11 \times (0.305)^2
$$
$$
(0.305)^2 = 0.093025
$$
$$
\text{Area} = 315.11 \times 0.093025 \approx 315.11 \times 0.093 \approx 29.305 \approx 29.3 \text{ sq meters}
$$

Wait, let's calculate more accurately:

$ 315.11 \times 0.093025 = 315.11 \times 0.093025 \approx 315.11 \times 0.09 = 28.3599 $, $ 315.11 \times 0.003025 \approx 0.953 $, total ≈ 28.3599 + 0.953 ≈ 29.313, so approximately 29.3 sq meters.

# Answer:
a) $\boxed{315}$ square feet (rounded to nearest square foot)
b) $\boxed{29.3}$ square metres (rounded appropriately)

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Calculate area of the rectangular part (excluding doors)

Step 2: Calculate area of the triangular part

Step 3: Calculate area of the circular window (subtract it)

Step 4: Total area of sheet metal

Step 1: Convert square feet to square meters

Answer: