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I am trying to minimize the cost of the fish tank. The cost of the tank is depending on the thickness of the tank.

  • The shape of tank is a cylinder without a top.
  • The tank is going to have a volume 250 gal.
  • If the tank height is greater then 24 in. then the thickness has to be 1/2 in. If the height is less or equal to 24 in., then the thickness can be 3/8 in..
  • Cost of 1/2 in. thick glass is 0.28 per sq. in..
  • Cost of 3/8 in. thick glass is 0.17 per sq. in..
Leucippus
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MikeJ
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1 Answers1

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If $r$ is the outer radius of the tank, $t$ is its thickness and $h$ is its height, the surface of the tank is $$2\pi rh+\pi r^2$$ The volumen is known, so you can write $$V=\pi (r-t)^2h$$ and solve this equation for $r$. You should express $V$ in cubic inches instead of gallons.

Substitute in the surface expression to build a function $S(h,t)$ which takes the height and the thickness and gives the surface.

Now define piecewise a function $f(h)$ that gives the cost: $$f(h)=\begin{cases} 0.17\cdot S(h,3/8)\text{ if }h\le24\\ 0.28\cdot S(h,1/2)\text{ if }h>24 \end{cases}$$

Differentiate $f$ to find its minimum. Note that the minimum could be at a discontinuity poit.

ajotatxe
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