Question

A rectangular pyramid has a base with sides 10 ft and 7 ft. The slant height where the base has the longer side is 7 ft, and the slant height where the base has the shorter side is 8 ft. Which of the following correctly calculates the surface area of the pyramid?

Answer 1

To calculate the surface area of a rectangular pyramid, we need to consider both the area of the base and the area of the triangular sides (lateral surface area).

1. **Calculate the area of the base**:
The base is a rectangle with lengths of 10 ft and 7 ft.
\[
\text{Area of base} = \text{length} \times \text{width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2
\]

2. **Calculate the area of the triangular sides**:
The pyramid has four triangular faces, two corresponding to each side of the base.

- **Triangles corresponding to the longer side (10 ft)**:
- The slant height for this side (which is the height of the triangle) is 7 ft.
- Each triangular face has a base of 10 ft.
- Area of one triangle = \( \frac{1}{2} \times \text{base} \times \text{height} \)
\[
\text{Area of one triangle} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]
- Total area for the two triangles on the longer side:
\[
2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2
\]

- **Triangles corresponding to the shorter side (7 ft)**:
- The slant height for this side is 8 ft.
- Each triangular face has a base of 7 ft.
\[
\text{Area of one triangle} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]
- Total area for the two triangles on the shorter side:
\[
2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2
\]

3. **Calculate total lateral surface area**:
\[
\text{Total lateral area} = \text{Area from longer side} + \text{Area from shorter side} = 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 126 \, \text{ft}^2
\]

4. **Calculate the total surface area of the pyramid**:
\[
\text{Surface area} = \text{Area of base} + \text{Total lateral area} = 70 \, \text{ft}^2 + 126 \, \text{ft}^2 = 196 \, \text{ft}^2
\]

Thus, the total surface area of the pyramid is \(\boxed{196 \, \text{ft}^2}\).