Optimization: Quiz
Problem 5
An open box constructed of material costing $2 per square meter is to have its length equal to twice its width. Find the dimensions of the box of largest volume than can be built with at most $30 of materials.
Answers, problem 5
Let L be the length, W be the width, and H be the height of the box, which is open on the top. The total surface area of the box is A = LW + 2HW + 2LH, since the top of the box is not included. Since the length is equal to twice the width, A = 2W2 + 6HW. The volume of the box is V = LWH = 2W2H.
The material to construct the box costs $2 per square meter, totaling at most $30, so the surface area is at most 15 square meters. Substituting this into the surface area formula, we can solve for one of the variables, say H:
H = (15 – W2)/3W
Therefore V = 2W2 (15 – W2)/3W = 10W – 2/3W3
To find the width that gives the maximum volume, we find dV/dt and set it equal to zero.
dV/dt = 0 = 10 – 2W2 => W = 51/2
Therefore L = 2(51/2)
Substitute both values into the surface area formula to get
H = 1/6(51/2)