To calculate how many gallons of paint are required to repaint the barn, we need to compute the total surface area to be painted and divide that by the coverage of one gallon of paint.

Surface Area Breakdown

The barn consists of:

1. Rectangular prism (lower part):

   • Four rectangular walls.
   • The dimensions provided are 84 feet in height and 38 feet in width.

2. Triangular prism (roof section):

   • Two triangular faces and two rectangular faces from the sides.

The four walls of the rectangular part and the two triangular faces need to be repainted.

Step 1: Rectangular Walls

There are two pairs of walls:

• Front and back walls (38 ft wide and 84 ft tall):
  Surface area per wall = $38 \times 84$ = 3,192 square feet
  Total area for the two walls = $3,192 \times 2$ = 6,384 square feet.

• Side walls (84 ft tall and depth from the diagram, 20 ft):
  Surface area per wall = $84 \times 20$ = 1,680 square feet
  Total area for the two walls = $1,680 \times 2$ = 3,360 square feet.

Total rectangular surface area = $6,384 + 3,360 = 9,744$ square feet.

Step 2: Triangular Faces

The height of the triangle is 12 ft, and the base is 38 ft.

The area of one triangular face is:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 38 \times 12 = 228 \text{ square feet}$ Total area for two triangular faces = $228 \times 2 = 456$ square feet.

Step 3: Total Surface Area to Paint

Total surface area = rectangular area + triangular area
$\text{Total surface area} = 9,744 + 456 = 10,200 \text{ square feet}$

Step 4: Calculating Gallons of Paint

Each gallon of paint covers 232 square feet. To find the number of gallons needed, divide the total surface area by the coverage per gallon:

$\text{Gallons required} = \frac{10,200}{232} \approx 43.97$

Since you can't buy a fraction of a gallon, you'll need 44 gallons of paint.

Would you like further clarification on any step?

Relative Questions:

1. How is the area of a triangular face calculated in geometry?
2. Why can't we use a fraction of a gallon in practical scenarios like painting?
3. How would you calculate the volume of the barn for storage capacity purposes?
4. What are the surface area formulas for prisms in general?
5. How could the total cost of paint be calculated if the price per gallon is provided?

Tip: Always round up when buying paint to ensure you have enough for touch-ups or unexpected areas.