The volume of an open-top rectangular box is 6700 in^3.

Question

The volume of an open top rectangular box is 6700in^3. The length of the rectangular base of the box is half the width. Find the dimensions so that the surface area is as small as possible

Answer 1

V = L•W•H and we know that L = 1/2W, so...

V = 1/2W•W•H

6700 = 1/2W²H ... solving for H ---------> H = 13,400/W²

Surface area is normally...

SA = 2WH + 2LH + 2LW... but in this case L = 1/2W and there is no lid so...

SA = 2WH + 2(1/2W)H + (1/2W)W... simplifying...

SA = 3WH + 1/2W²

Next you can substitute H in terms of W that we solved above in the surface area (so it's one variable)...

SA = 3W(13,400/W²) + 1/2W² ... simplifying

SA = 40,200/W + 1/2W²

We want to minimize surface area (goal of the question), so you graph this monstrosity on your TI-84 (surface area as a function of width) and simply look for a local minimum in the 1st quadrant (assuming this rectangular box has positive measurements).

A little calculator skill will show you that a width of about 34.256 inches results in a minimum surface area of about 1,760.254 square inches (you don't need to know this number - only that it's a minimum). The length is half of the width. The height we solved as a function of width above - just plug it in. Therefore...

L ≈ 17.128 inches

W ≈ 34.256 inches

H ≈ 11.419 inches