Question

A silo (base not included ) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is 5 times as great fit the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed at 8,000 cubic units and the cost of construction is to be kept at minimum. Neglect the thickness of the silo and waste in construction.
I have the function of the cost as
C=(16000/r) + (26pi r^2 /3)
It is now asking for the interval of interest. ..in interval notation. I can see that it can't be zero so I have (0,__] , but I have no idea how to find end point on the interval of interest.
Please please please help!

Answer 1

We are given the equation:

C = (16,000 / r) + (26πr^2 / 3)

First get the 1st derivative:

dC/dr = (-16,000 / r^2) + (52π r / 3)

Equate dC/dr to get the minima:

(-16,000 / r^2) + (52π r / 3) = 0

Multiply everything by r^2:

-16,000 + (52π/3) r^3 = 0

r = 6.65

 

Therefore the cost at radius = 6.65 is:

C = (16,000 / 6.65) + (26π (6.65)^2 / 3)

C = 3,610.07

So perhaps the interval interest is (0, 3,610.07] if cost or (0, 6.65] if based on radius.