math 171 question 3

Question

The area of a rectangular piece of cardstock is 93.5 square inches. The cardstock is used to make an open-top box by cutting 1 inch squares off of each corner and turning up the sides. If the box has a volume of 58.5 cubic inches, find the length and width of the cardstock used. Assume that the length is greater than the width

Answer 1

93.5 = Area before 1 inch corners are cut off and the sides turned up into a box

58.5 = volume of the box = 1 in.by width by length = 1wL=wL

58.5 = wL

w =58.5/L

(w+2)(L+2) = 93.5

wL + 2L +2w + 4 = 93.5

58.5 + 2L +2w = 89.5

2L +2w= 89.5-58.5 = 31

L+w = 31/2 = 15.5

L+58.5/L= 15.5

L^2-15.5L +58.5 = 0

2L^2 -31L +117 = 0

use the quadratic formula to solve for L

L = 31/4 +/-.25sqr(31^2-8(117))

L =7.75 +/-.25sqr25

= 7.75. +/-1.25

= 9 or 6.5 inches

w=58.5/9 or 58.5/6.5 =6.5 or 9 inches

Length = 9 inches

width = 6 1/2 inches

Volume = 1 inch by 9 inches by 6 1/2 inches = 1(9)(6.5) = 58.5 cubic inches

Area = (9+2)(6.5+2) = 11(8.5) = 93.5 square inches