To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the four triangular faces.
-
Area of the base: The base is a rectangle. \[ \text{Area of the base} = l \times w = 5.5 , \text{yd} \times 4 , \text{yd} = 22 , \text{yd}^2 \]
-
Area of the triangular faces: There are four triangular faces, two with a base of \( l \) and height from the apex to the midpoint of \( l \), and two with a base of \( w \) and height from the apex to the midpoint of \( w \).
First, we need to find the slant heights of the triangular faces.
For the slant height \( s \):
-
When the base is \( l = 5.5 , \text{yd} \): \[ \text{slant height} = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{\left(\frac{5.5}{2}\right)^2 + (6)^2} = \sqrt{(2.75)^2 + 36} = \sqrt{7.5625 + 36} = \sqrt{43.5625} \approx 6.6 , \text{yd} \]
-
When the base is \( w = 4 , \text{yd} \): \[ \text{slant height} = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{\left(\frac{4}{2}\right)^2 + (6)^2} = \sqrt{(2)^2 + 36} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32 , \text{yd} \]
-
-
Area of each triangular face:
-
The area of the triangular faces with base \( l \): \[ \text{Area} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 5.5 \times 6.6 \approx 18.15 , \text{yd}^2 \] Since there are two of these faces: \[ \text{Total area} \text{ for } l = 2 \times 18.15 \approx 36.30 , \text{yd}^2 \]
-
The area of the triangular faces with base \( w \): \[ \text{Area} = \frac{1}{2} \times b \times h = \frac{1}{2} \times 4 \times 6.32 \approx 12.64 , \text{yd}^2 \] Since there are two of these faces: \[ \text{Total area} \text{ for } w = 2 \times 12.64 \approx 25.28 , \text{yd}^2 \]
-
-
Total surface area: \[ \text{Total surface area} = \text{Base area} + \text{Total area of triangular faces} = 22 + 36.30 + 25.28 \approx 83.58 , \text{yd}^2 \]
Based on the provided response options, none of them correspond to the calculated area. However, the closest option to the derived calculation after verification might yield a slight approximation error in case of estimation and rounding.
Among the provided options, 83.19 yd² is the closest response but may not precisely match the calculated surface area.
If choosing among the response options, one may select: 83.19 yd².