Day 3: Optimizing Surface Area and Volume of a Cylinder
1. What are the dimensions of the rectangular prism that minimizes its surface area AND
maximizes its volume?
2. What dimensions should a cylinder have in order to minimize its surface area AND
maximize its volume? Explain your reasoning.
3. How could you verify or disprove your hypothesis in #2?
Investigation 1:
Maximize Volume of a Cylinder for a Given Surface Area
Your task is to design a cylindrical juice can that uses 470cm2 of aluminum. The can
should have the greatest volume possible.
1) Substitute the given surface area and radius into the surface area formula and
solve for the height, h.
2) Determine the volume of this can using the formula for the volume of a cylinder:
V  r 2 h . Record the data in the following table:
Surface Area
470
470
470
Radius
2
5
8
Height
Volume
3) What is the maximum volume for the cans in your table? What are the radius and
height for this can?
5. Do these dimensions give the optimal volume for the surface area of 470cm2? Explain
your answer.
To determine the radius of the can with a given surface area, use the surface area formula and
substitute h = 2r.
Given:
SA = 2πr2 + 2πrh and h = 2r
SA = 2πr2 + 2πr(2r)
Simplify and Isolate r:
Investigation 2:
Minimize Surface Area for a Given Volume
Your task is to design a cylinder with a volume of 400cm3 using the smallest
amount of metal possible.
1.
2.
For each radius measurement, calculate the area of the base of the
can.
Using the formula Vcylinder  area of base height  , substitute the volume (400) and the
area of the base (calculated in #1). Solve for the height.
3.
Radius
2
4
6
Calculate the total surface area of the cylinder, including the top and bottom.
Area of Base
Volume
400
400
400
Height
Surface Area
4.
Compare the surface areas and dimensions of the cylinders. Choose the cylinder that
has the least surface area. How does its height compare to its radius?
5.
Are these dimensions the optimal ones? Explain.
To determine the radius of the can with a given volume, use the volume formula and substitute
h = 2r.
Given:
V = πr2h and h = 2r
V = πr2(2r)
Simplify and Isolate r:
Consolidate:
To maximize the volume for a given surface area, the optimal cylinder should just fit inside a
____________ with side length equal to _______________.
This means that the height of the cylinder should be ______________.
To maximize the volume, the radius = _____________________.
To minimize the surface area for a given volume, the optimal cylinder should also just fit inside
a ____________ with side length equal to _______________.
This means that the height of the cylinder should be ______________ .
To minimize the surface area, the radius = ___________________.