calculus optimization see below

Question

An observatory is to be in the form of a right circular cylinder surmounted by a hemispherical dome. If the hemispherical dome costs 22 times as much per square foot as the cylindrical wall, what are the most economic dimensions for a volume of 10000 cubic feet?

Answer 1

Hi Abigail.

 

How about I give you some hints to get you started. You will need formulas for the volume of a right circular cylinder and a sphere. You will also need a formula for the surface area of those objects. The problem does not state whether the "floor" of the observatory is included (and that makes a difference).

 

The V= 10000 is there so that you can establish a relationship between r (radius of both objects) and h (the height of the cylinder). Remember the top is a "hemisphere" so you need to take 1/2. Set 10000 equal to the sum of the volumes of the cylinder and the hemisphere and solve for h. Now when you create the Cost function you will be able to substitute for h in terms of r. Then find the derivative of the cost function with respect to r. Set it equal to zero and solve for r. You should probably try to convince yourself that that value of r is generating a minimum cost (2nd derivative test?).

 

What does the cost function look like?

 

The surface area of a sphere is 4 pi r², so for the hemisphere it will be 1/2 of that. Also the dome costs 22 times more that the cylinder material.

 

Cost of dome = 2 pi r².

 

Cost of cylinder (not considering the floor) = 2pi r h

 

So, cost function: C(x) = 44 pi r² + 2 pi r h  (multiply the dome cost by 22).

 

Now substitute for h, find C'(x) with respect to r, etc...