Question

for exercises 7 and 8, determine the surface areas of the composite figures. 7. 8.

Explanation:

Step1: Analyze figure 7

For the first composite - figure, we consider the surface - area of each part and subtract the hidden areas. First, find the surface area of the larger rectangular prism and the smaller one separately and then adjust for the common face.
The larger rectangular prism has dimensions \(l_1=(4 + 3)=7\) ft, \(w_1 = 8\) ft, \(h_1=9\) ft. The surface area of a rectangular prism \(S = 2(lw+lh + wh)\). So, \(S_1=2(7\times8 + 7\times9+8\times9)=2(56 + 63+72)=2\times191 = 382\) square feet.
The smaller rectangular prism has dimensions \(l_2 = 8\) ft, \(w_2 = 4\) ft, \(h_2 = 3\) ft. Its surface area \(S_2=2(8\times4 + 8\times3+4\times3)=2(32 + 24 + 12)=2\times68 = 136\) square feet.
The hidden area (the common face) has an area of \(A_h=8\times4 = 32\) square feet.
The surface area of the composite figure 7 \(S_{7}=S_1+S_2 - 2A_h=382+136-2\times32=382 + 136 - 64=454\) square feet.

Step2: Analyze figure 8

For the second composite - figure, the larger rectangular prism has dimensions \(l = 8\) in, \(w = 8\) in, \(h = 20\) in. Its surface area \(S_{l}=2(8\times8 + 8\times20+8\times20)=2(64 + 160+160)=2\times384 = 768\) square inches.
The smaller rectangular prism has dimensions \(l = 4\) in, \(w = 4\) in, \(h = 5\) in. Its surface area \(S_{s}=2(4\times4 + 4\times5+4\times5)=2(16 + 20+20)=2\times56 = 112\) square inches.
The hidden area (the common face) has an area of \(A_{h1}=4\times4 = 16\) square inches.
The surface area of the composite figure 8 \(S_{8}=S_{l}+S_{s}-2A_{h1}=768+112 - 2\times16=768+112 - 32=848\) square inches.

Answer:

The surface area of figure 7 is 454 square feet and the surface area of figure 8 is 848 square inches.