Question

calculating surface area of a composite figure

choose the answers that make each sentence true.

the surface area of the triangular prism without the shared base is (square) (ft^{2}).

the surface area of the rectangular prism without the shared base is (square) (ft^{2}).

if the exterior of the composite figure is being painted, the area to be painted is approximately (square) (ft^{2}).

Explanation:

Step1: Calculate the surface area of the triangular prism without the shared base

The triangular prism has two triangular faces and two rectangular faces.
The area of a triangular face is \(A_{triangle}=\frac{1}{2}\times base\times height\). Here, base = 12 ft and height = 6 ft. So \(A_{triangle}=\frac{1}{2}\times12\times6 = 36\space ft^{2}\). There are two triangular faces, so \(2\times36=72\space ft^{2}\).
The area of a rectangular face with dimensions 8.5 ft and 60 ft is \(A_{rect1}=8.5\times60 = 510\space ft^{2}\). There are two such rectangular faces, so \(2\times510 = 1020\space ft^{2}\).
The total surface area of the triangular prism without the shared base is \(72+1020=1092\space ft^{2}\).

Step2: Calculate the surface area of the rectangular prism without the shared base

The rectangular prism (excluding the shared base). The formula for the surface - area of a rectangular prism \(S = 2(lw+lh+wh)\) (but we exclude one face).
The dimensions of the rectangular prism: length \(l = 60\) ft, width \(w=12\) ft, height \(h = 12\) ft.
The surface area without the shared base: \(S=(2\times(12\times60))+(2\times(12\times12))+(12\times60)\)
\(S = 1440+288 + 720\)
\(S=2448\space ft^{2}\)

Step3: Calculate the total area to be painted

The total area to be painted is the sum of the surface area of the triangular prism (without shared base) and the surface area of the rectangular prism (without shared base).
\(A=1092 + 2448=3540\space ft^{2}\)

Answer:

The surface area of the triangular prism without the shared base is \(1092\space ft^{2}\).
The surface area of the rectangular prism without the shared base is \(2448\space ft^{2}\).
The area to be painted is \(3540\space ft^{2}\).