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Question goes...A can is in the shape of a circular cylinder is required to have a volume of 750 cubic centimeters. The top and bottom are made of material that costs 8 cents per square centimeter while the sides are made of a material that costs 5 cents per square centimeter. Express the total cost $C$ of the material as a function of the radius $r$ of the cylinder. For what value of $r$ is the cost $C$ the least?

Here's what I did but for some reason I'm not too confident with it.

Top and bottom: 2 circles; area of circle is $\pi r^2$, so total area is $2\pi r^2$.

Side: Lateral surface area is $2\pi rh$.

$C=2\pi r^2(.08)+2\pi rh(.05)$.

$V=\pi r^2h$

$750=\pi r^2h$

$h=\frac{750}{\pi r^2}$

So now $C=2\pi r^2(.08)+2\pi r\left(\frac{750}{\pi r^2}\right)$

$C=.16\pi r^2+.1\pi r\left(\frac{750}{\pi r^2}\right)$

$C=.16\pi r^2+\frac{75}{r}$ (multiply $16\pi r^2$ by $r$ to get same denominator)

$C=\frac{.16\pi r^3+75}{r}$.

I put this in my graph and got the minimum to be \$4.21 but that seems too high to be an answer.

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Please consider accepting some of the answers you where given at your previous questions (by clicking the tick). –  Rasmus Sep 21 '11 at 20:53
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You lost the factor $0.05$ in the line "So now..." –  TMM Sep 21 '11 at 20:55
    
Oh sorry, Ill do that right now. and that was a typing error I have the .05 on my paper, but am i right? –  Ronnie.j Sep 21 '11 at 21:01
    
Congratulations on making the "reality check" of whether answer makes sense. It doesn't, but $8$ cents per square cm is awfully expensive, unless the stuff is gold-plated. –  André Nicolas Sep 21 '11 at 23:02

1 Answer 1

In order to prove your answer you need to find first derivative of $C(r)$ and than calculate $r$ from equation $C'_r=0$ (in that point function $C(r)$ has minimum)

$C'_r=\frac{(0.16\pi r^3+75)'r-r'(0.16\pi r^3+75)}{r^2}\Rightarrow C'_r=\frac{0.32\pi r^3-75}{r^2}$ , so $C'_r=0 \Rightarrow 0.32\pi r^3-75=0 \Rightarrow r^3=\frac{75}{0.32\pi} \Rightarrow r=\sqrt[3]{\frac{75}{0.32\pi}}\Rightarrow r=4.21 cm$ , And if we substitute this value into $C(r)$ we get minimal cost of can which is $26.72$ cents.

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That can't be right -- for 26.72 cents, you get less than $6 cm^2$ of the cheaper material... –  TonyK Oct 24 '11 at 15:18

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