Question

(a) find the following side lengths for the net.

a = ft

b = ft

c = ft

d = ft

(b) use the net to find the surface area of the prism.

Explanation:

Step1: Identify side lengths from the prism and its net

• The prism has a triangular base with sides \(3\) ft, \(4\) ft and a base of \(5\) ft, and height of the prism (rectangular part) is \(6\) ft.
• For the net:
• \(A\) is the length of one of the non - base sides of the triangular base. So \(A = 4\) ft.
• \(B\) is the height of the prism (length of the rectangle). So \(B=6\) ft.
• \(C\) is the length of the other non - base side of the triangular base. So \(C = 3\) ft.
• \(D\) is the base of the triangular base. So \(D = 5\) ft.

Step2: Calculate the surface area of the prism using the net

The surface area of a triangular prism \(S=2\times(\text{Area of the triangular base})+\text{Perimeter of the triangular base}\times\text{Height of the prism}\)

• Area of the triangular base \(A_{triangle}=\frac{1}{2}\times3\times4 = 6\) square feet.
• Perimeter of the triangular base \(P=3 + 4+5=12\) feet.
• Height of the prism \(h = 6\) feet.
• \(S=2\times6+12\times6\)
• First, calculate \(2\times6=12\) and \(12\times6 = 72\)
• Then \(S=12 + 72\)

Answer:

(a) \(A = 4\) ft, \(B = 6\) ft, \(C = 3\) ft, \(D = 5\) ft
(b) \(84\) square feet