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 find the surface area of a rectangular pyramid, we need to consider both the area of the base and the areas of the triangular faces.

### Given:
- Base dimensions: 10 ft (longer side) and 7 ft (shorter side)
- Slant heights: 7 ft (for the side with length 10 ft) and 8 ft (for the side with length 7 ft)

### Step 1: Calculate the area of the base.
The area \( A_{\text{base}} \) of the rectangular base is given by:

\[
A_{\text{base}} = \text{length} \times \text{width} = 10 \, \text{ft} \times 7 \, \text{ft} = 70 \, \text{ft}^2
\]

### Step 2: Calculate the area of the triangular faces.
There are four triangular faces corresponding to the sides of the base:

1. **Triangular Face corresponding to the 10 ft side (Height = 7 ft)**:
- Base of triangle = 10 ft
- Height of triangle = 7 ft

The area \( A_1 \) of this triangle is given by:

\[
A_1 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]

2. **Triangular Face corresponding to the 7 ft side (Height = 8 ft)**:
- Base of triangle = 7 ft
- Height of triangle = 8 ft

The area \( A_2 \) of this triangle is given by:

\[
A_2 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]

Since there are two triangular faces corresponding to each dimension:

- Two triangular faces with base = 10 ft:
\[
A_{10} = 2 \times A_1 = 2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2
\]

- Two triangular faces with base = 7 ft:
\[
A_{7} = 2 \times A_2 = 2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2
\]

### Step 3: Calculate total surface area.
Now, we sum the area of the base and the areas of the triangular faces:

\[
\text{Total Surface Area} = A_{\text{base}} + A_{10} + A_{7}
\]

\[
\text{Total Surface Area} = 70 \, \text{ft}^2 + 70 \, \text{ft}^2 + 56 \, \text{ft}^2 = 196 \, \text{ft}^2
\]

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