SOLUTION: An open water tank with a square base is to be made from a thin sheet of metal. Show that the following are the dimensions so that the least amount of metal is used to make a tank

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Question 1137439: An open water tank with a square base is to be made from a thin sheet of metal. Show that the following are the dimensions so that the least amount of metal is used to make a tank of capacity 8m^3.
Height of the tank: 1.26m
Length of the square base: 2.52m
Thanks :)
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!

Given: We have a square prism (aka block) and we want this prism to have a volume of 8 cubic meters.

x = side of the square base
h = height
Volume of prism = length*width*height
Volume of prism = x*x*h
Volume of prism = x^2*h
x^2*h = volume
x^2*h = 8

Now isolate h by dividing both sides by x^2
x^2*h = 8
h = 8/(x^2)
This will be used later

The surface area of the tank involves adding up the areas of the five sides that make up the open water tank. Keep in mind that there is no top for this square prism.
The bottom face is x*x = x^2 in area
One side face is x*h, and there are four of them, so they total 4xh for the lateral surface area.
Overall, the total surface area of this tank is x^2+4xh which represents the amout of sheet metal we use. The goal is to get this as small as possible.

Now plug in h = 8/(x^2)
SA = x^2+4xh
SA = x^2+4x(8/(x^2)) ... note how the h term is now gone
SA = x^2+(32/x)
SA = (x^3)/x+(32/x)
SA = (x^3+32)/x
f(x) = (x^3+32)/x

So the goal is to minimize f(x)

From here we would use Calculus or a graphing calculator to determine the local min. Assuming you aren't in calculus, I'll go with the second option. Here is the graph of f(x) = (x^3+32)/x

(note: ignore the part where x is negative. A negative length makes no sense)
Use the "min" feature on your calculator to locate the local min to be roughly located at (2.52, 19.05) which is shown as point A in the diagram above.

If you dont have a TI 83 or TI84 graphing calculator, then you can use this free tool to help graph. After typing (x^3+32)/x into the box, you should get this graph. The neat thing about that tool is that you can click on the lowest point and have the coordinates pop up (note: you may need to click two times). This is what you should see on your screen after clicking the lowest point

This lowest point (2.52, 19.05) means that the surface area is minimized when the side length is x = 2.52 meters approximately. This lowest surface area is approximately y = 19.05 square meters, which is the smallest amount of metal we can use to form a 8 cubic meter tank.

Use this approximate x value to find the height
h = 8/(x^2)
h = 8/((2.52)^2)
h = 8/6.3504
h = 1.25976316452507
h = 1.26

So that explains how your teacher got the answers
height = 1.26 meters
length of square base = 2.52 meters