Question

Find the minimum cost of the material for a rectangular container with a square base that contains 16 cubic feet if the material for the top and bottom cost $2 per square foot and the material for the sides cost $1 per square foot.

Please show work for me thank you so much!

Answer 1

What have you learned so far? Have you learned to use Lagrange Multipliers?

Answer 2

No i have not, these are just considered optimization problems.

Answer 3

Is this a calculus 1 optimization or is this one using partial derivatives? (Sorry, these questions matter. No use in taking a hard way if you don't need to.)

Answer 4

It is the one using partial derivatives, and i understand completely don't be sorry you are being so kind by helping me out! I know that you must find the derivative and then set it equal to zero to find the critical point, but I am just very confused on how to find it since we have only done 2 in class so far.

Answer 5

(Then I have to put the obligatory statement that this is not the true min value. But the result will suffice for class as of now. Once you get to Lagrange multipliers, do it again on this problem and you will see the results differ ;) but that's not the point now...)

Since you have a rectangular prism with square bases, you have 4 rectangles and 2 squares. Let x denote the length of the squares. Consequently, I will assign y to the width of the rectangles.
The area of 1 rectangle is xy; the area of 1 square is x^2. Thus, the surface area is
SA = 4xy + 2x^2
and consequently, the cost is:
C = 4x^2 + 4xy
and the volume is
V = (x^2)*y = 16

Now you solve the volume equation for y:
y = 16/(x^2)
And substitute it back into the cost equation and you optimize. It is peculiar though that you do not have a z component. Thus you do nt need to incur the partials.

Answer 6

Thank you so much!