Question

which expression can be used to find the surface area of the trapezoidal prism? what is the surface area of the trapezoidal prism? s.a. = 162 ft² s.a. = 6·3 + 6·4 + 6·5 + 6·8 + 23·4 + 1/2(3·4) = 18 + 24 + 30 + 48 + 36

Explanation:

Step1: Identify surface - area formula parts

The surface - area of a trapezoidal prism is the sum of the areas of all its faces. The formula used here seems to be the sum of the areas of the rectangular faces and the trapezoidal faces. The areas of the rectangular faces are \(6\times3\), \(6\times4\), \(6\times5\), \(6\times8\). The area of the trapezoidal faces is calculated as \(2\times[\frac{1}{2}(3 + 4)\times3]\).

$$SA=6\times3+6\times4+6\times5+6\times8 + 2\times[\frac{1}{2}(3 + 4)\times3]$$

Step2: Calculate each part

First, calculate the areas of the rectangular faces:

• \(6\times3 = 18\)
• \(6\times4=24\)
• \(6\times5 = 30\)
• \(6\times8=48\)

Then, calculate the area of the trapezoidal faces:

$$2\times[\frac{1}{2}(3 + 4)\times3]=2\times[\frac{1}{2}\times7\times3]=21$$

Step3: Sum up all areas

$$SA=18 + 24+30 + 48+21=141$$

(There seems to be an error in the given answer of 162 in the picture. The correct calculation based on the formula shown is as above). But if we follow the calculation in the picture without checking its correctness in terms of the prism's structure:

$$SA=6\times3+6\times4+6\times5+6\times8+2\times[\frac{1}{2}(3\times4)]$$

$$=18 + 24+30 + 48+12$$

$$=132$$

(This is also different from 162. Assuming the correct expression is \(SA = 6\times3+6\times4+6\times5+6\times8+2\times[\frac{1}{2}(3\times4)]\))

Answer:

If we assume the correct expression for surface - area calculation is \(6\times3+6\times4+6\times5+6\times8+2\times[\frac{1}{2}(3\times4)]\), the surface - area is \(132\) square feet. If we calculate it in a more correct way considering trapezoid area formula \(A=\frac{1}{2}(a + b)h\) for trapezoidal faces, we get \(141\) square feet. There is likely an error in the provided answer of \(162\) square feet in the original image.