Question

A farmer is building a new cylindrical silo with a flat roof and an earthen floor that will hold `20000 m^3` of corn. What dimensions of the silo will minimize the materials required for construction?

Answer 1

I tried this:

Explanation:

Have a look:

Answer 2

Numerically, to three significant figures, `r=h=18.5 m`.

Explanation:

The question asks us to minimise the surface area of the cylinder ignoring the floor.

If the radius of the silo is `r`, then the area of the circular roof is `pir^2`. The area of the circular wall is its height times its circumference: `2pirh`. So the surface area of the construction is `2pirh+pir^2=pir(2h+r)`.

The volume of the silo is given as 20000 `m^3`, and is the area of the base times the silo's height: `pir^2h`. If we set this to the given amount, we can derive an expression for `h` in terms of `r` that we can substitute in to our surface area expression:

`pir^2h=20000`
`h=20000/(pir^2)`

So the surface area is
`pir(2*20000/(pir^2)+r)=40000/r+pir^2`.

As this is now a function only of `r`, we can find its extrema.
Let `A(r)=40000/r+pir^2`. Then `(dA)/(dr)=-40000/r^2+2pir`.

`(dA)/(dr)=0rArr-40000/r^2+2pir=0rArr2pir^3=40000`
`r^3=20000/pirArrr=root(3)(20000/pi)=10root(3)(20/pi)`

So the function has a single real root, the other two being complex conjugates of each other. Now let us deduce its nature via inspection of the second derivative.

`(d^2A)/(dr^2)=80000/r^3+2pi`

`(d^2A)/(dr^2)(10root(3)(20/pi))=80000pi/20000+2pi=4pi+2pi=6pi>0`

This the construction surface area is minimum at the single extremum.

Thus the desired dimensions are:
Radius, `r=10root(3)(20/pi)`
Height, `h=20000/(pir^2)=20000/(pi*100root(3)(400/pi^2))=200/root(3)(400pi)=100/root(3)(50pi)=10root(3)(20/pi)`, which is the same expression as for the radius. So the needed height and the needed radius are the same.

Numerically, to three significant figures, `r=h=18.5 m`.

Sanity check the solution by plotting out the graph of radius vs. surface area:
graph{y=40000/x + pi x^2 [10, 30, 3000, 4500]}