Question

practice
the net of a rectangular prism is shown below. what is the total surface area of the prism in square inches?
a. $472\\ \text{in.}^2$
b. $400\\ \text{in.}^2$
c. $328\\ \text{in.}^2$
d. $192\\ \text{in.}^2$

Explanation:

Step1: Identify dimensions

From the net, the dimensions of the rectangular prism are length \( l = 16 \) in, width \( w = 8 \) in, and height \( h = 4.5 \) in (wait, no, looking at the net, maybe I misread. Wait, the middle square? Wait, no, the net of a rectangular prism has 6 faces: 2 of each pair (length×width, length×height, width×height). Wait, looking at the diagram, the vertical rectangle has height 16 in and width 8 in? Wait, no, the small side: the height of the small rectangle is \( 4\frac{1}{2} = 4.5 \) in? Wait, maybe the dimensions are length \( l = 16 \) in, width \( w = 8 \) in, height \( h = 4.5 \) in? Wait, no, let's re-examine. Wait, the net: the central rectangle (the base) and then the other faces. Wait, actually, the formula for the surface area of a rectangular prism is \( SA = 2(lw + lh + wh) \). Let's find the correct dimensions. Wait, maybe the length is 16 in, width is 8 in, and height is \( 4.5 \) in? Wait, no, maybe I made a mistake. Wait, looking at the options, let's check the numbers. Wait, maybe the height is 4.5? Wait, no, 4.5 is \( 4\frac{1}{2} \). Wait, let's compute with \( l = 16 \), \( w = 8 \), \( h = 4.5 \). Then \( lw = 16×8 = 128 \), \( lh = 16×4.5 = 72 \), \( wh = 8×4.5 = 36 \). Then \( SA = 2(128 + 72 + 36) = 2(236) = 472 \)? No, that's option A. Wait, but maybe the dimensions are different. Wait, maybe the length is 16, width is 8, and height is 4? No, the diagram has \( 4\frac{1}{2} \). Wait, maybe I misread the height. Wait, the small rectangle's height is \( 4\frac{1}{2} \) in, the length of the horizontal rectangles is 16 in, and the vertical rectangle's width is 8 in. So the three dimensions are \( l = 16 \) in, \( w = 8 \) in, \( h = 4.5 \) in. Then surface area: \( 2(lw + lh + wh) \). Let's calculate each term:

• \( lw = 16×8 = 128 \)
• \( lh = 16×4.5 = 72 \)
• \( wh = 8×4.5 = 36 \)

Now sum these: \( 128 + 72 + 36 = 236 \). Then multiply by 2: \( 2×236 = 472 \). So the surface area is 472 in², which is option A.

Step2: Verify the formula

The formula for the surface area of a rectangular prism is indeed \( SA = 2(lw + lh + wh) \), where \( l \) is length, \( w \) is width, and \( h \) is height. We identified the dimensions from the net, calculated each pair of faces, summed them, and multiplied by 2 to get the total surface area.

Answer:

A. \( 472 \text{ in}^2 \)