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)

Question

Responses

202 ft.2
202 ft. squared

70 ft.2
70 ft. squared

196 ft.2
196 ft. squared

126 ft.2

Answer 1

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

1. Area of the base: The base is a rectangle, so its area can be calculated as: \[ \text{Area}_{\text{base}} = \text{length} \times \text{width} = 10 , \text{ft} \times 7 , \text{ft} = 70 , \text{ft}^2 \]

2. Area of the triangular faces:

   • There are two triangular faces with a base of 10 ft (the longer side).

   • The area of one of these triangles can be calculated using the slant height of 7 ft: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 , \text{ft} \times 7 , \text{ft} = 35 , \text{ft}^2 \]

   • Since there are two such triangles: \[ \text{Total Area}_{\text{longer side triangles}} = 2 \times 35 , \text{ft}^2 = 70 , \text{ft}^2 \]

   • Now for the two triangular faces with a base of 7 ft and slant height of 8 ft: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 , \text{ft} \times 8 , \text{ft} = 28 , \text{ft}^2 \]

   • Since there are two such triangles: \[ \text{Total Area}_{\text{shorter side triangles}} = 2 \times 28 , \text{ft}^2 = 56 , \text{ft}^2 \]

3. Total surface area: \[ \text{Surface Area} = \text{Area}{\text{base}} + \text{Total Area}{\text{longer side triangles}} + \text{Total Area}_{\text{shorter side triangles}} = 70 , \text{ft}^2 + 70 , \text{ft}^2 + 56 , \text{ft}^2 = 196 , \text{ft}^2 \]

Thus, the surface area of the pyramid is 196 ft². Therefore, the correct answer is:

196 ft.²