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

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


   

Question 1183388: An open lid tank to be made by concrete has width 50𝑐𝑚, inside capacity of 4000 𝑚3
and square base. Find the inner dimension of the tank with the minimum volume of
concrete.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The thickness of the open-lid tank is 50 cm = %281%2F2%29+m. Let the interior square base have a side of length +x+ meters.
Also let the inner height be y+ m.
This then gives
x%5E2%2Ay+=+4000
since it is given that the inside capacity is 4000+m%5E3.
For the exterior of the tank, the square base has a side of length
+x+%2B+2%2A%281%2F2%29+=+x%2B1 meters, and a height of y+%2B+1%2F2 meters.
The external volume is then
V+=+%28x%2B1%29%5E2%2A%28y%2B1%2F2%29+=+%28x%2B1%29%5E2%2A%28+4000%2Fx%5E2+%2B1%2F2%29, after substituting for y from the previous equation.
==> ,
after simplifying the expression on the right side.
Setting dV%2Fdx+=+%28%28x%2B1%29%2A%28x%5E3+-+8000%29%29%2Fx%5E3+=+0, we get x+=+20+m.
We cannot accept x+=+-1 since the domain of V is x+%3E+0.
Now if x+%3C+20, dV%2Fdx+%3C+0+, and
when x+%3E+20, dV%2Fdx+%3E+0+, so by the the 1st derivative test,
there is a local minimum at x+=+20. Since it is the only critical
point in the domain (0, infinity), the minimum at x+=+20 is also
absolute minimum.
Therefore the inner dimensions of the tank are 20 m x 20 m x 10m. (The height is
y+=+4000%2Fx%5E2+=+4000%2F20%5E2+=+10+m.)