1. Square pieces of cardboard will be cut out of the corners of a 32 inch by 40
ID: 2882980 • Letter: 1
Question
1. Square pieces of cardboard will be cut out of the corners of a 32 inch by 40 inch piece of cardboard, then the sides will be folded up to create an open-top box. Find the maximum volume of the box and the dimensions
2. A box will have a square base and no top. The material used to construct the base costs $0.50 per square foot, while the material for the sides costs $0.30 per sqaure foot. If the box must have a volume of 18 cubic feet, then find the minimum cost and the dimensions that minimize the cost.
Explanation / Answer
1. length of acrd board = 40
width = 32
square of side = x
Vol = x * ( 40 -2x) ( 32-2x) = 4x^3 - 144x^2 +1280x
dV/dx = 12x^2 - 288 x + 1280
d^2V/dx^2 = 24x - 288
dV /dx = 0
12x^2- 288x + 1280 = 0
3x^2 - 72x+ 32 = 0
x = 23.5 or x = 0.5
Max V = 2467.5 at x = 23.5
2. V= 18
18 = a^3
a = (18)^1/3 = 2.62
Min cost = 2.62*2.62*0.50 +4*2.62*2.62*0.30 = 2.62*2.62( 0.50+1.20) = 11.66$
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