Question

a box designer has been given the task of determining the surface area of various open boxes (no lids) that can be constructed by cutting four equal sized square corners from a 15.4-in by 11.5-in sheet of cardboard and folding up the sides. surface area is the total area of the boxes sides and bottoms. to find a formula that relates the total surface areas, s, measured in square inches of the open box to the size of the square cutouts, x, measured in inches. use function notation

Answer 1

Since we will cut 4 squares of side x from each corner, then the dimensions will less by 2x (x from each side

The box will have 3 dimensions

L, W, H

Since the dimensions of the cardboard are 15.4 and 11.5 inches

We will subtract each one by 2x to find L, and W

L = 15.4 - 2x

W = 11.5 - 2x

H = x

We will find the area of the base

Area of base = L * W

Area of base = (15.4 - 2x)(11.5 - 2x)

There are two opposite faces with dimensions L, H

The area of these two faces = 2[(15.4 - 2x)(x)] = 2x(15.4 - 2x)

There are two opposite faces with dimensions W, H

The area of these two faces = 2[(11.5 - 2x)(x)] = 2x(11.5 - 2x)

The surface area of the box is the sum of the area of the 5 faces

S.A = (15.4 - 2x)(11.5 - 2x) + 2x(15.4 - 2x) + 2x(11.5 - 2x)

Let us simplify it

S.A = 177.1 - 53.8x + 4x^2 + 30.8x - 4x^2 + 23x - 4x^2

Let us add the like terms

S.A = 177.1 - 4x^2