Create a box from a piece of cardboard that is 12 inches by 8 inches by cutting equal squares out of the corners. Express the volume of the box of a function of x.

Question

1. What is the domain of your function?
2. What is V(1)?V(2)?V(3)?V(4)?
3. What is the maximum value of box assuming the size of the cut is an integer?
4. What size cut out gives a volume of 12 cubic inches?

Answer 1

V=x(12-2x)(8-2x) = x(96-40x +4x^2) = 96x-40x^2 + 4x^3

V' = 12x^2 -80x + 96 = 0

3x^2 -20x + 24=0

x = 20/6 + or - (1/6)sqr(400-288) = 10/3 + or - (1/6)sqr112 = 3 1/3 + or - 2sqr28 = 3 1/3 + or -4sq7

x = 3.33 + or - 2.65 = 5.98 or 1.38, ignore 5.98 as it 8-2x makes one side negative

x = 1.38 for maximum volume

for an integer use x=1

V = (12-2)(8-2) = 60

if x=2 then V = (12-4)(8-4)2 = 8(4)2 = 64

if x=3 V=(12-6)(8-6)3 = 36

if x =4 V =0

max V is when x=2

domain is x=0, 1, 2, 3 and 4

V=12 only if x is not an integer

solve 12=96x-40x^2 + 4x^3, solve for x

should get 2 positive solutions, one a fraction between 0 and 1 and the other between 3 and 4

one should be slightly more than 1/8 and the other slightly less than 3 3/4 for volume = 12