Shin C. answered • 04/25/20
UCLA Alumni | AP Calculus AB/BC & College Calculus Specialist
Hello M T! I'll help you out!
Imagine you have a rectangular prism now rotated sideways (so that the square base is on the bottom and top, and it should be long length in the vertical). If it's still confusing, try to draw it out based on what I just stated. Call the lengths of the base x, and the length of the vertical length y.
Because we know V = 68 and that V = Length * Width * Height = (x ^ 2) * y = 68. For now, rearrange this to get y = 68 / (x ^ 2).
The top and bottom bases are both squares, so the area of each of those would be x ^ 2. Additionally, the rectangular faces that would be on the side would have area x * y. Because there are four of them, it would become 4xy.
Using the earlier-rearranged equation, substitute that to get 4 * x * ( 68 / x ^ 2 ) = 272 / x.
In this problem, we can say that the cost (in dollars) = area of each part * rate (in dollars / sq ft).
Therefore, we can make the cost equation (note how I added all three areas with the respective rates).
C(x) = (x ^ 2) * 0.2 + (x ^ 2) * 0.1 + (272 / x) * 0.025 = 0.3 * (x ^ 2) + 6.8 / x
Minimum cost will occur IF C ' (x) = 0 and/or does not exist AND C ' (x) changes from negative to positive.
C ' (x) = 0.6x - 6.8 * ( x ^ (-2) )
The only time when C ' (x) does not exist is when x = 0 (because 6.8 / 0 is undefined). But that does not matter because we wouldn't care if the area of the base was 0 * 0 (that's useless). Ignore that.
Now, let's check when C ' (x) = 0. That would be when:
0 = 0.6x - 6.8 * ( x ^ (-2) ) ⇒ 6.8 * ( x ^ (-2) ) = 0.6x ⇒ 6.8 / 0.6 = x ^ 3 ⇒ x = 2.25.
Checking sign of C ' (x) does indeed tell us that at x < 2.25, C ' (x) < 0, and x > 2.25, C ' (x) > 0.
Therefore, x = 2.25 and y = 68 / (2.25 ^ 2) = 13.43 yield the minimum cost of the box! >>> (ANSWER)
Hope this helped you! Feel free to ask me additional questions!
