Question

Beth is moving into a new house and purchased cardboard boxes for packing her things. Each cardboard box has a square base and a height that is 5 inches shorter than the side length of the base. If x represents the side length of the base, which of the following functions represents the surface area, S, of the cardboard box?
S(x) = 2x2 + 10x
S(x) = 6x2 + 20x
S(x) = 2x2 - 10x
S(x) = 6x2 - 20x

Answer 1

Answer: Choice D)
S(x) = 6x^2 - 20x

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Explanation:

x = side length of base
x*x = x^2 = area of base

The top also has an area of x^2 since the base and top are both congruent squares. The total base area is x^2+x^2 = 2x^2

The height h is 5 inches shorter than the base, so
h = (base length) - 5
h = x-5

Each lateral side is of area h*x = (x-5)*x = x^2-5x

There are 4 lateral sides
Total lateral area = 4*(area of one lateral side)
Total lateral area = 4*(x^2-5x)
Total lateral area = 4*x^2-4*5x
Total lateral area = 4*x^2-20x

Add the total lateral area (4x^2-20x) to the total base area (2x^2)

Doing so gets us
S(x) = Total Surface Area
S(x) = (Area of bases) + (area of lateral sides)
S(x) = (2x^2) + (4x^2-20x)
S(x) = (2x^2+4x^2) - 20x
S(x) = 6x^2 - 20x
which is why the answer is choice D