For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders.

Question

For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 39 in.³ and is constructed using the least amount of metal.

Let r and h (in in.) be the radius and height of the container, respectively.

r =  in. (Round your answer to two decimal places, if necessary.)

Complete the following parts.

(a)    Give a function f in the variable r for the quantity to be optimized.

f(r) = 

 

(b)    State the domain of this function. (Enter your answer using interval notation.)

 

(c)     Give the formula for h in terms of r.

Answer 1

Let's make the assumption the the thickness of the material is the same for all portions of the right cylinder. "Constructed using the least amount of metal" then means to minimize the surface area. So, step 1 is to write an equation for the surface area using the variables given ("r" and "h").

SA = surface area of top and bottom + surface area of sides

SA = 2(πr²) + (2πrh)

We are given information that the volume of the can MUST be 39 in³ and volume is πr²h so

V = πr²h

39 = πr²h

h = 39/(πr²)

We can now plug in "39/πr²" in place of "h" in our surface area equation:

SA = 2πr² + 2πr*(39/πr²)

SA(r) = 2πr² + 78/r

SA(r) = 2πr² + 78r^-1

Or, as requested in one of the questions, f(r) = 2πr² + 78r^-1

To minimize, take the derivative and set it equal to find the minimum:

SA'(r) = 4πr - 78r^-2 (using the power rule)

SA'(r) = 4πr - 78/r²

0 = 4πr - 78/r²

0 = 2πr³/r² - 39/r²

0 = 1/r²(2πr³ - 39)

1/r² cannot contribute to making this expression equal zero although critical points also occur when the derivative is not defined. However, in this case, that would be when r = 0 which is not an answer that would result in a can for the stew. So we can ignore the 1/r²:

2πr³ - 39 = 0

2πr³ = 39

r³ = 39/(2π)

r = cuberoot of (39/(2π)) ≈ 1.84 inches

The domain would be the smallest "r" and the largest r. The smallest r would of course be something just slightly over zero but that would make the height really tall. So, if you think about the tallest can they could put on a shelf, maybe a height of 12 inches?? which would equate to a radius of about 1 inches. To find the largest r, you could imagine a huge "r" with an extremely small "h". How huge? Totally depends on the manufacturing process or perhaps the store shelves. I'll let you answer that. Maybe 6 inches or so ????

As far as a function of "h" in terms of "r", we did that with h(r) = 39/(πr²)