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
I tried this:
Explanation:
Have a look:
Numerically, to three significant figures,
Explanation:
The question asks us to minimise the surface area of the cylinder ignoring the floor.
If the radius of the silo is
The volume of the silo is given as 20000
So the surface area is
As this is now a function only of
Let
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.
This the construction surface area is minimum at the single extremum.
Thus the desired dimensions are:
Radius,
Height,
Numerically, to three significant figures,
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]}