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?

2 Answers
Jun 21, 2018

I tried this:

Explanation:

Have a look:
enter image source here

Jun 21, 2018

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]}