Differentiation: An Optimization Problem

PROJECT: Marking Term 3
Chapter 3: Applications of Derivatives
tuna

Is Something Fishy? An Optimization Problem

Corporations are always trying to save money and increase profits. Waste leads to smaller profits. So, you would think they would want to use the minimum amount of material in packaging their items.

Part I
A typical can of tuna fish has a height of 3.7 cm and a diameter of 8.5 cm (assume the can is a perfect right cylindrical solid).

a. Find the volume of the can in cubic centimeters.

b. Create a formula for the surface area of the can as a function in terms of height and radius.

c. Use the volume formula and solve for height in terms of radius.

d. Using the expression for height from part c, rewrite the formula for surface area so that it is function in terms of radius.

e. Use differentiation to find the radius of a can that yields minimum surface area. State the dimensions of the cylinder with minimal surface area for this particular volume (Pt.I, a).

f. How does the "typical can of tuna fish" compare with the "ideal" cylinder with minimal surface area?

A tin can container, such as a tuna can, is not a solid cylinder. The container is really more of a thin skin of material (aluminum or tin or whatever) holding in the contents. If the can were cut open we would see it was comprised of two cylinderical discs (top and bottom) and a sheet for the lateral sides. These have a thickness and thus the material of the can has a volume.

Part II
Again using a typical can of tuna fish with height of 3.7 cm and a diameter of 8.5 cm:

a. Assume the material of the actual can has a constant thickness of 0.01 cm. Use this "thickness" and your formula for the surface area of the can as a function of its radius from Part I, d to create a function of the Volume of material comprising the actual can for this specific capacity.

b. Use differentiation to find the radius of a can that yields minimum volume. State the dimensions of the cylinder with minimal surface area for this particular capacity.

c. Confirm that your radius from part b yields a minimum volume of material by creating a table of values for several possible radii.

d. How does the "typical can of tuna fish" compare with the "ideal" cylinder with minimal material volume?

e. Explain why the material volume can be minimized, but that there is no maximum volume of material that could be used in the can construction. [HINT: Use either the second derivative or limits in your explanation.]

This page was created on March 8, 2004.
Last Updated on August 25, 2004.

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a.	Find the volume of the can in cubic centimeters.
b.	Create a formula for the surface area of the can as a function in terms of height and radius.
c.	Use the volume formula and solve for height in terms of radius.
d.	Using the expression for height from part c, rewrite the formula for surface area so that it is function in terms of radius.
e.	Use differentiation to find the radius of a can that yields minimum surface area. State the dimensions of the cylinder with minimal surface area for this particular volume (Pt.I, a).
f.	How does the "typical can of tuna fish" compare with the "ideal" cylinder with minimal surface area?

a.	Assume the material of the actual can has a constant thickness of 0.01 cm. Use this "thickness" and your formula for the surface area of the can as a function of its radius from Part I, d to create a function of the Volume of material comprising the actual can for this specific capacity.
b.	Use differentiation to find the radius of a can that yields minimum volume. State the dimensions of the cylinder with minimal surface area for this particular capacity.
c.	Confirm that your radius from part b yields a minimum volume of material by creating a table of values for several possible radii.
d.	How does the "typical can of tuna fish" compare with the "ideal" cylinder with minimal material volume?
e.	Explain why the material volume can be minimized, but that there is no maximum volume of material that could be used in the can construction. [HINT: Use either the second derivative or limits in your explanation.]

<<< Back to Mr. Whitney's Mathematics Home Page	Site Created & Maintained by
	Matthew C. Whitney