To determine the surface area of a prism with the given parameters, we need to follow these steps:
1. Calculate the Lateral Surface Area:
The lateral surface area of a prism is found by multiplying the perimeter of the base by the height of the prism. Given the parameters:
- Perimeter of the base (P) = 80
- Height (H) = 4
The formula for the lateral surface area (LSA) is:
[tex]\[
\text{Lateral Surface Area} = \text{Perimeter of the base} \times \text{Height} = 80 \times 4 = 320
\][/tex]
2. Calculate the Area of One Base:
Since the base is rectangular, we can find the area using the dimensions of the base. Given the parameters:
- One side of the base (length) = 16
- The perimeter of the base = 80
The perimeter of a rectangle is given by:
[tex]\[
\text{Perimeter} = 2 \times (\text{Length} + \text{Width})
\][/tex]
Solving for the width (W):
[tex]\[
80 = 2 \times (16 + W)
\][/tex]
[tex]\[
80 = 32 + 2W
\][/tex]
[tex]\[
48 = 2W
\][/tex]
[tex]\[
W = 24
\][/tex]
The area of the base (A) is:
[tex]\[
\text{Area of the base} = \text{Length} \times \text{Width} = 16 \times 24 = 320
\][/tex]
3. Calculate the Total Surface Area:
The total surface area of the prism is the sum of the lateral surface area and the area of the two bases.
- Lateral Surface Area = 320
- Area of one base = 320
- Since there are two bases, the combined area of the bases = 2 \times 320 = 640
The formula for the total surface area (TSA) is:
[tex]\[
\text{Total Surface Area} = \text{Lateral Surface Area} + \text{Area of the two bases} = 320 + 640 = 960
\][/tex]
Therefore, the surface area for the prism is:
[tex]\[
\text{Lateral Surface Area} = 320
\][/tex]
[tex]\[
\text{Area of the base} = 320
\][/tex]
[tex]\[
\text{Total Surface Area} = 960
\][/tex]
