Question

a cake is created by assembling two layers as shown in the diagram below. the exposed surfaces of the cake require frosting. assuming the 4 inch by 10 inch base on which the cake sits will not be frosted, what is the area that will require frosting? 10 in. 2 in. 6 in. 6 in. 4 in. 10 in. 352 square inches 372 square inches 392 square inches 432 square inches

Explanation:

Step1: Calculate the surface area of the bottom layer

The bottom layer is a rectangular prism with length \(l = 10\) in, width \(w=4\) in, and height \(h = 6\) in.
The surface area of a rectangular prism is \(2(lw+lh + wh)\). But we don't frost the base (\(lw\) part).
The area of the bottom - layer (excluding the base): \(lw+2lh + 2wh\)
Substitute \(l = 10\), \(w = 4\), \(h = 6\):
\(10\times4+2\times10\times6+2\times4\times6\)
\(=40 + 120+48\)
\(=208\) square inches.

Step2: Calculate the surface area of the top layer

The top layer is a rectangular prism with length \(l = 10\) in, width \(w = 2\) in, and height \(h=6\) in. But we don't frost the part that is in contact with the bottom layer (\(10\times2\) area).
The area of the top - layer: \(2lh+2wh\) (since the top surface of the top - layer is exposed)
Substitute \(l = 10\), \(w = 2\), \(h = 6\):
\(2\times10\times6+2\times2\times6+10\times2\)
\(=120 + 24+20\)
\(=164\) square inches.

Step3: Sum up the two areas

Add the areas of the two layers: \(208+164=372\) square inches.

Answer:

372 square inches