SOLUTION: An open lid tank to be made by concrete has width 50𝑐𝑚, inside capacity of 4000𝑚3 and square base. Find the inner dimension of the tank with the minimum volume of concret

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Question 1182974: An open lid tank to be made by concrete has width 50𝑐𝑚, inside capacity of 4000𝑚3 and square base. Find the inner dimension of the tank with the minimum volume of concrete.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the inside length in meters of a side of the square base; let h be the height/depth of the tank.

Then the volume of the tank is

x%5E2h=4000 [1]

Since the thickness of the concrete is 50cm = 0.5m, the square base has dimensions (x+1)(x+1)(0.5).

The four sides of the tank can be viewed as four congruent rectangular solids each with dimensions (x+0.5)(h)(0.5).

So the total volume of the tank is

V=0.5%28x%2B1%29%5E2%2B4%28x%2B0.5%29%28h%29%280.5%29 [2]

Solve [1] for h in terms of x and substitute in the volume formula to get the volume in terms of the single variable x.

h+=+4000%2Fx%5E2

V=0.5%28x%2B1%29%5E2%2B4%28x%2B0.5%29%284000%2Fx%5E2%29%280.5%29
V=0.5%28x%2B1%29%5E2%2B%282x%2B1%29%284000%2Fx%5E2%29
V=0.5%28x%2B1%29%5E2%2B8000%2Fx%2B4000%2Fx%5E2

Differentiate and set the derivative equal to zero to find the length of the side of the square base that minimizes the volume of concrete.

dV%2Fdx=%28x%2B1%29-8000%2Fx%5E2-8000%2Fx%5E3

x%5E3%28x%2B1%29-8000x-8000+=+0
x%5E3%28x%2B1%29-8000%28x%2B1%29+=+0
%28x%5E3-8000%29%28x%2B1%29+=+0

x%5E3=8000 or x=-1

Clearly the negative solution makes no sense in the problem. So

x%5E3=8000
x+=+20

ANSWER: The volume of concrete to make the tank is minimized when the side length of the square base is 20m