Question

a right triangular prism and its net are shown below. (all lengths are in feet.) (a) find the following side lengths for the net. ( a=square mathrm{ft} ) ( b=square mathrm{ft} ) ( c=square mathrm{ft} ) ( d=square mathrm{ft} ) (b) use the net to find the surface area of the prism. ( square mathrm{ft}^{2} )

Explanation:

Step1: Identify side lengths

• For \(A\): In the right - triangular prism, \(A\) corresponds to the side length of \(4\) ft (one of the legs of the right - triangle base).
• For \(B\): \(B\) is the height of the prism, which is \(6\) ft.
• For \(C\): \(C\) corresponds to the side length of \(3\) ft (the other leg of the right - triangle base).
• For \(D\): \(D\) is the hypotenuse of the right - triangle base. Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 3\) and \(b = 4\), we have \(c=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\) ft.

Step2: Calculate surface area

The surface area \(S\) of a right - triangular prism is given by \(S=2\times(\text{area of base})+\text{lateral surface area}\).

• Area of the triangular base \(A_{base}=\frac{1}{2}\times3\times4 = 6\) \(ft^{2}\).
• Lateral surface area: The lateral faces have areas \(3\times6\), \(4\times6\), and \(5\times6\). The sum of lateral face areas \(A_{lateral}=(3 + 4+5)\times6=12\times6 = 72\) \(ft^{2}\).
• Then \(S=2\times6+72\).

$$S = 12+72=84$$

\(ft^{2}\)

Answer:

a) \(A = 4\) ft, \(B = 6\) ft, \(C = 3\) ft, \(D = 5\) ft
b) \(84\) \(ft^{2}\)