An open box is to be made from a flat square piece of material 12 inches in length and width by cutting equal squares of length x x from the corners and folding up the sides.

Question

(a) Write the volume VV of the box as a function of xx. Leave it as a product of factors; you do not have to multiply out the factors.

(b) Give the domain of this volume function in interval notation

Answer 1

square 12 inches by 12 inches

cut square corners x by x from each of the 4 corners

fold them up creating a box that's x by (12-2x) by (12-2x) = volume= x(12-2x)(12-2x) = x(144-48x+4x^2) = 4x^3-48x^2 + 144x.

Volume = V(x) = x(12-2x)^2

for volume to be >0, x must be less than 6 inches.

0<x<6

max V is when V'=0, when 12x^2-96x+144=0

when x^2 -8x+12=0

(x-6)(x-2)=0

x=2 or 6. x=6 means V=0

x=2 means V=2(12-4)^2 = 2(8^2) =2(64)=128

0<V<128 in^3

domain for Volume in interval notation is (0, 128]

Volume is a non-negative amount, no greater than 128 cubic inches