The problem describes a rectangular prism (cake) with height $h = 4$ inches and base dimensions $l = 12$ inches and $w = 15$ inches. We need to find the surface area of the part of the cake that will have frosting, which includes the four sides and the top.

1. Problem Description

The problem describes a rectangular prism (cake) with height $h = 4$ inches and base dimensions $l = 12$ inches and $w = 15$ inches. We need to find the surface area of the part of the cake that will have frosting, which includes the four sides and the top.

2. Solution Steps

First, calculate the area of the top of the cake, which is a rectangle with sides $l$ and $w$:

$A_{top} = l \times w = 12 \times 15 = 180$ square inches.

Next, calculate the area of the four sides. The two sides with length $l$ have area $l \times h$, and the two sides with length $w$ have area $w \times h$. Thus, the total area of the four sides is:

$A_{sides} = 2(l \times h) + 2(w \times h) = 2(12 \times 4) + 2(15 \times 4) = 2(48) + 2(60) = 96 + 120 = 216$ square inches.

The total surface area to be frosted is the sum of the area of the top and the area of the four sides:

$A_{total} = A_{top} + A_{sides} = 180 + 216 = 396$ square inches.

3. Final Answer

396