Question
An open-top cylinder is to be constructed from a sheet of steel. If we ask that the cylinder has a volume of 10 cubic metres, what are the dimension of the cylinder which requires the least amount of material to make?
Solution, part 1
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Before reading the solution, you should really try the problem for yourself. By should I mean that if you don't you will not gain any understanding, whatever you might believe, this is 100% guaranteed. So do yourself a favour, try the problem and look at the solution once you've gave it an authentic try.
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Solution, part 2
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Before reading the solution, you should really try the problem for yourself. By should I mean that if you don't you will not gain any understanding, whatever you might believe, this is 100% guaranteed. So do yourself a favour, try the problem and look at the solution once you've gave it an authentic try.
Solution, part 2
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To find the minimum value of the function
![{\displaystyle A(r)=\pi r^{2}+{\frac {20}{r}}\qquad {\text{for }}r>0}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d06cfbe7034f8234f4966a637356c9c115f11c12)
We first compute its derivative
![{\displaystyle A'(r)=2\pi r-{\frac {20}{r^{2}}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/58a4830407fbd6ddf5ac9375b99b677cc7f85126)
And then solve it for zero to find its critical points
![{\displaystyle 2\pi r-{\frac {20}{r^{2}}}=0\quad \iff \quad \pi r^{3}=10\quad \iff \quad r={\sqrt[{3}]{\frac {10}{\pi }}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/2c9f9e48120a4e8f738a36f1335e12228c0c48b3)
We'll use the second derivative test to make sure this critical point is the minimum value of the function.
![{\displaystyle A''(r)=2\pi +{\frac {40}{r^{3}}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/3b935614aa972595b9305fab091cdc93f0a8e6fc)
And clearly, this function is positive for any value of which is positive, so we can guarantee the critical point we found is a minimum.
Substituting the exact value back to find shows that we get
![{\displaystyle \displaystyle {h=r={\sqrt[{3}]{\frac {10}{\pi }}}}\approx 1.47}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/6de53197604ac19599f680cc3bdf39051d3091de)
So the dimensions of the cylinder using the least amount of material has its radius equal its height.
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