calculus homework problem from

Question

having trouble figuring out my homework problem from calculus book:  a right circular cylinder can hold 24 cubic inches of water while the cost for the material for the bottom and top of the can is $0.02 per square inch and the cost for the sides is $0.01 per square inch. Find the radius that will minimize the cost. Use your surface area formula of s=2(pi)(r)(h)+2(pi)r^2) and volume formulas v=pir^2h for your calculations.

Answer 1

You are being asked to find the radius that will minimize the cost, so your goal will be to write the cost as a function of solely r, and then use the first derivative to find a minimum.

 

From the surface area equation, which is already broken into the two pieces, one of which represents the area of the side and the other which represents the area of the top and bottom, we can multiply each of those pieces by their respective cost per square inch to find a cost function.

 

Therefore Cost=(.01 dollars per square inch) * 2(pi)(r)(h) + (.02 dollars per square inch) * 2(pi)(r²).

 

This function has more than one variable, and we want it to be written only in terms of r, therefore we need to eliminate h from this equation.  We were given the total volume of the can, which can be used with the volume function to express h in terms of r.

 

V=(pi)(r²)(h)

24 = (pi)(r²)(h)

24 / ((pi)(r²)) = h

 

Now we can substitute this into the cost function to obtain a cost function containing only r as a variable:

Cost = (.48) / r  + (.04)(pi)(r²)

 

Taking the first derivative and setting equal to zero to find the minimum, we have:

0 = (-.48) / r² + (.08)(pi)(r)

 

Rearranging to solve for r:

r = cube root( (.48) / ((.08)(pi)) )

r = 1.24 cm

 

Therefore the minimum cost will occur when the radius is 1.24 cm.