Dimensions

Question

(a) You have been asked to bid on the construction of a square-bottomed box with no top
which will hold 100 cubic inches of water. If the bottom and sides are made from the same
material, what are the dimensions of the box which uses the least material? (Assume that
no material is wasted.)
(b) Suppose the box in part (a) uses different materials for the bottom and the sides. If the
bottom material costs 5¢ per square inch and the side material costs 3¢ per square inch,
what are the di

Answer 1

first, assign variables to the box dimensions:

L = width

D = depth

 

equations based on the description:

 

V = volume = L^2 * D= 100 cubic inches

 

Question a) asks about the total amount of material, so we need an equation for the total area of sides and bottom:

M = L^2 + 4LD  (each side is L * D)

 

We need this equation to have only one variable on the right, so replace D with 100 / L^2:

 

M = L^2 + 4L (100 / L^2)

 

M = L^2 + 400 / L

 

Now---we can find the minimum in one of 3 ways:

--graphing

--calculus

--trial and error

 

Here is the calculus method:

M = f(L) = L^2 + 400L^-1

 

differentiate:

dM/dL = 2L - 400L^-2

 

for minimum M, the derivative will be 0, so

2L = 400/L^2

L^3 = 400

L = 7.37 inches

D = 100 / 7.37^2 = 1.841 inches

M = L^2 + 4LD = 108.56 square inches

 

Let's test this--try L slightly smaller and larger than 7.37:

Smaller:

L = 7.36

D = 100 / 7.36^2 = 1.85

M = 108.64  (M is higher)

 

Larger:

L = 7.38

D = 100 / 7.38^2 = 1.836

M = 108.66  (M is also higher)

 

so--it looks like we did find the right minimum