Question

for exercises 7 and 8, determine the surface areas of the composite figures. 7. 3 ft 8 ft 4 ft 3 ft 9 ft 8. 4 in. 4 in. 5 in. 20 in. 8 in. 8 in.

Explanation:

Exercise 7

Step1: Analyze the individual - part dimensions

We have two rectangular - prisms. The bottom prism has dimensions \(l_1 = 8\) ft, \(w_1 = 4\) ft, \(h_1 = 3\) ft. The top prism has dimensions \(l_2=4 + 3=7\) ft, \(w_2 = 4\) ft, \(h_2 = 9 - 3 = 6\) ft.

Step2: Calculate the surface - area of the bottom prism

The surface - area formula for a rectangular prism is \(SA = 2(lw+lh + wh)\). For the bottom prism, \(SA_1=2(8\times4 + 8\times3+4\times3)=2(32 + 24+12)=2\times68 = 136\) square feet.

Step3: Calculate the surface - area of the top prism

For the top prism, \(SA_2=2(7\times4+7\times6 + 4\times6)=2(28 + 42+24)=2\times94 = 188\) square feet.

Step4: Subtract the overlapping area

The overlapping area is \(2\times(4\times4)=32\) square feet.

Step5: Calculate the total surface area

The total surface area \(SA = SA_1+SA_2-32=136 + 188-32=292\) square feet.

Exercise 8

Step1: Analyze the individual - part dimensions

We have two rectangular - prisms. The bottom prism has dimensions \(l_1 = 8\) in, \(w_1 = 8\) in, \(h_1 = 20\) in. The top prism has dimensions \(l_2 = 4\) in, \(w_2 = 4\) in, \(h_2 = 5\) in.

Step2: Calculate the surface - area of the bottom prism

Using the formula \(SA = 2(lw+lh + wh)\), for the bottom prism, \(SA_1=2(8\times8+8\times20 + 8\times20)=2(64 + 160+160)=2\times384 = 768\) square inches.

Step3: Calculate the surface - area of the top prism

For the top prism, \(SA_2=2(4\times4+4\times5 + 4\times5)=2(16 + 20+20)=2\times56 = 112\) square inches.

Step4: Subtract the overlapping area

The overlapping area is \(2\times(4\times4)=32\) square inches.

Step5: Calculate the total surface area

The total surface area \(SA=SA_1 + SA_2-32=768+112 - 32=848\) square inches.

Answer:

Exercise 7: 292 square feet
Exercise 8: 848 square inches