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?(1 point)
Responses

202 ft2
202 ft squared

70 ft2
70 ft squared

196 ft2
196 ft squared

126 ft2

Answer 1

To calculate the surface area of the rectangular pyramid, we need to find the area of the base and the areas of the triangular faces.

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

2. **Side Areas:**
There are four triangular faces. We will calculate the area of each triangular face next.

- **Triangle on longer side (10 ft)**:
- Base = 10 ft
- Height = 7 ft (slant height corresponding to the longer side)
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{ft} \times 7 \, \text{ft} = 35 \, \text{ft}^2
\]

- **Triangle on shorter side (7 ft)**:
- Base = 7 ft
- Height = 8 ft (slant height corresponding to the shorter side)
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \, \text{ft} \times 8 \, \text{ft} = 28 \, \text{ft}^2
\]

Therefore, the surface area includes:
- Two triangles with 10 ft base area: \( 2 \times 35 \, \text{ft}^2 = 70 \, \text{ft}^2 \)
- Two triangles with 7 ft base area: \( 2 \times 28 \, \text{ft}^2 = 56 \, \text{ft}^2 \)

3. **Total Surface Area Calculation:**
\[
\text{Total Surface Area} = \text{Base Area} + \text{Side Areas}
= 70 \, \text{ft}^2 + 70 \, \text{ft}^2 + 56 \, \text{ft}^2
= 70 \, \text{ft}^2 + 126 \, \text{ft}^2
= 196 \, \text{ft}^2
\]

Thus, the correct calculation for the surface area of the pyramid is:
\[
\boxed{196 \, \text{ft}^2}
\]

So the correct response from the options is **196 ft²** or **196 ft squared**.