Word Problem

Question

A rectangular bin with an open top and volume of 30 cubic feet is to be built. The length of its base must be three times he width and the bin must be at least 7 feet high. Material for the base of the bin costs $9 per square foot and material for the sides costs $6 per square foot. If it costs $500 to build the bin, what are its dimensions?

Answer 1

Let L = 3w be length, w = width of base, and H  = height.  It helps to draw a picture & label length, width, height.

 

V = LWH = 3w²H = 30  so  H = 10/w² ≥ 7  or  (taking reciprocal & reversing inequality symbol) w² ≤ 10/7; w ≤ 1.195.

 

Cost based on surface area = 9•3w² + 7•(2Hw + 2•3HW) = 500 where first term represents the one base @ $9/sq. ft. and the next component is at $7/sq. ft. and consists of two ends & two long sides (front & back).

 

Now what to do?  Why not use the cost equation to solve for H as a function of w? Then make a table of H values, based on w running from 0.1 (very skinny) up to 1.2.  I did so & observed that the H values ran from very high (c. 89 ft high with w = 0.1 down to H(1) = 8.45 & H(1.05) = 8, H(1.15) = 7.21 and finally H(1.2) = 6.86 (the latter violating the minimum height requirement of 7 ft.

 

I guess that, from a practical standpoint, w should = 1.195 ft, based on the earlier inequality step.

 

Comments welcome!