Expert: Ahmed Salami - 12/8/2008

Question
QUESTION: This comes from a sample final for my final on Monday, Dec. 8. Thanks.

A rectangular box with volume 18ft^3 is to be built with a square base and
NO top. the material used for the bottom panel costs $2 per ft^2 while the
material used for the side panels costs $1.50 per ft^2. Find the minimum
cost of such a box. Justify your answer using the methods of calculus.

Thanks for you time and help. It's extremely appreciated.

ANSWER: Hi Chelsea,
Let the box have a base length of b and a height h, the volume V is therefore
V = b^2.h = 18
The area of the bottom panel is b^2 and its cost is then
b^2
There are four sides each with an area of bh, its cost is therefore
(1.5).4.bh = 6bh
The total cost of the box is
C = b^2 + 6bh
From b^2.h = 18,
h = 18/b^2
Therefore,
C = b^2 + 6b.18/b^2
 = b^2 + 108/b
The minimum value of C occurs when dC/db = 0 and d^2C/db^2 > 0
dC/db = 2b - 108/b^2
equating to zero,
2b - 108/b^2 = 0
2b = 108/b^2
b = 54/b^2
b^3 = 54

That is the value of b that gives the minimum cost, use it to get the cost. Please cross-check, did this in kind of a haste because it was urgent for you and i was busy.

Regards



---------- FOLLOW-UP ----------

QUESTION: Thank you for your help!!! :D
I just have one question....
would the cost of the bottom panel be: 2b^2?

Answer
Hi Chelsea,
I regret the error. Yes, the cost of the bottom panel is 2b^2 which means a revise to the solution.
C = 2b^2 + 6bh
From b^2.h = 18,
h = 18/b^2
Therefore,
C = 2b^2 + 6b.18/b^2
 = 2b^2 + 108/b
The minimum value of C occurs when dC/db = 0 and d^2C/db^2 > 0
dC/db = 4b - 108/b^2
equating to zero,
4b - 108/b^2 = 0
4b = 108/b^2
b = 27/b^2
b^3 = 27
b = 3ft

d^2C/db^2 = 4 - -2(108)/b^3
         = 4 + 216/b^3
which is positive at b = 3, ensuring that the cost is a minimum.
The minimum cost is therefore,
C = 2(3^2) + 108/3
 = 2(9) + 36
 = $54.00

I'm sorry for the mistake.

Regards.