A relief agency plans to have special-purpose cylindrical barrels made for shipping large quantities of grain. Each barrel is to contain 2 cubic meters of grain. The material to be used for the sides of the cylinder costs $5 per meters squared, while the material for the two ends costs $10 per meters squared, due to extra strength requirements. Determine the minimum costs of materials to manufacture each barrel and the dimensions required to attain this minimum cost.
Here's my work:
\(\displaystyle \
\L\
\begin{array}{l}
v = \pi r^2 h \to 2 = \pi r^2 h \to h = \frac{2}{{\pi r^2 }} \\
A = 2\pi r^2 + 2\pi rh \\
= 10(2\pi r^2 ) + 5(2\pi r\frac{2}{{\pi r^2 }}) \\
= 20\pi r^2 + \frac{{20}}{r} \\
\frac{{dA}}{{dx}} = 40\pi r - \frac{{20}}{{r^2 }} \\
= \frac{{40\pi r^3 - 20}}{{r^2 }} \\
0 = \frac{{40\pi r^3 - 20}}{{r^2 }} \\
= 40\pi r^3 - 20 \\
= 20(2\pi r^3 - 1) \\
= 2\pi r^3 - 1 \\
r^3 = \frac{1}{{2\pi }} \\
r = \frac{1}{{\sqrt[3]{{2\pi }}}} \\
\end{array}
\\)
I get lost after that point. I can't figure out how to solve for h (r is too complex).
My book has an answer here for the minimum cost which I have no idea how they got (I know they used the equation for the surface area and pluged in the value of r, assuming they have the same value of r I got). Extrema problems are really bumming me out.
\(\displaystyle \
\L\
c = 20\sqrt[3]{{2\pi }} + 20\pi \frac{1}{{2\pi ^{\frac{2}{3}} }}
\\)
One other quesiton, how do you find the domain for this?
Here's my work:
\(\displaystyle \
\L\
\begin{array}{l}
v = \pi r^2 h \to 2 = \pi r^2 h \to h = \frac{2}{{\pi r^2 }} \\
A = 2\pi r^2 + 2\pi rh \\
= 10(2\pi r^2 ) + 5(2\pi r\frac{2}{{\pi r^2 }}) \\
= 20\pi r^2 + \frac{{20}}{r} \\
\frac{{dA}}{{dx}} = 40\pi r - \frac{{20}}{{r^2 }} \\
= \frac{{40\pi r^3 - 20}}{{r^2 }} \\
0 = \frac{{40\pi r^3 - 20}}{{r^2 }} \\
= 40\pi r^3 - 20 \\
= 20(2\pi r^3 - 1) \\
= 2\pi r^3 - 1 \\
r^3 = \frac{1}{{2\pi }} \\
r = \frac{1}{{\sqrt[3]{{2\pi }}}} \\
\end{array}
\\)
I get lost after that point. I can't figure out how to solve for h (r is too complex).
My book has an answer here for the minimum cost which I have no idea how they got (I know they used the equation for the surface area and pluged in the value of r, assuming they have the same value of r I got). Extrema problems are really bumming me out.
\(\displaystyle \
\L\
c = 20\sqrt[3]{{2\pi }} + 20\pi \frac{1}{{2\pi ^{\frac{2}{3}} }}
\\)
One other quesiton, how do you find the domain for this?
