Final answer:
The surface area of a regular pentagonal prism with a height of 3.5 inches and a base edge length of 2 inches is approximately 38.44 square inches.
Step-by-step explanation:
To find the surface area of a regular pentagonal prism, we need to calculate the areas of the two pentagonal bases and the five rectangular faces that connect them. For a regular pentagon with a side length s, the formula for the area of one pentagonal base (using the approximate value of the pentagonal constant, τ ≈ 1.72048) can be given by:
Base Area = τ * s² / 4
For the sides, since we have a regular pentagon, there will be five rectangles with a height (h) of 3.5 inches and a width (w) equal to the side length of the pentagon, which is 2 inches:
Lateral Area = 5 * h * w
Now, we compute these values:
Base Area = 1.72048 * (2 in)² / 4 ≈ 1.72048 * 4 in² / 4 ≈ 1.72048 in²
Lateral Area = 5 * 3.5 in * 2 in = 5 * 7 in² = 35 in²
The total surface area is the sum of the area of the two bases and the lateral area:
Total Surface Area = 2 * Base Area + Lateral Area
Total Surface Area = 2 * 1.72048 in² + 35 in² ≈ 2 * 1.72048 in² + 35 in² ≈ 3.44096 in² + 35 in² ≈ 38.44096 in²
Round the result to the nearest hundredth,
Total Surface Area ≈ 38.44 in²