Question

Arlan needs to create a box from a piece of cardboard. The dimensions of his cardboard are 10 inches by 8 inches. He must cut a square from each corner of the cardboard, in order to form a box. What size square should he cut from each corner, in order to create a box with the largest possible volume?
A. 0.5 inches
B. 1 inch
C. 1.5 inches
D. 2 inches

Answer 1

1) Dimensiones of the cardboard:

length: 10 inches
width: 8 inches

2) dimensions of the squares cut

length: x
width: x

3) dimensions of the box:

length of the base = 10 - 2x
width of the base  = 8 - 2x
height = x

4) Volume of the box


V = (10 - 2x) (8 - 2x) x = x [80 - 20x - 16x + 4x^2] = x [ 80 - 36x + 4x^2 ] =

V = 80x - 36x^2 + 4x^3

5) Maximum volume => derivative of V, V' = 0

V' = 80 - 72x + 12x^2 = 0

6) Solve the equation

Divide by 4 => 3x^2 - 18x + 20 = 0

Use the quadratic formula: x = 1.47 and x = 4.53 (this is not valid)

So, the answer is the option C. 1.5 inches.