Cleo Cola, a new Fortune 500 company, plans to sell soda pop in 12-ounce cylindrical cans. Cleo Cola will generously reward the employee team that designs the can that can be constructed with the least amount of material.
Your team is challenged to determine the dimensions that will minimize the surface area of the can. (Your team knows that 12 liquid ounces is approximately equal to 355 cubic centimeters.) Assume that the thickness of the material is uniform (i.e., the thickness of the aluminum is the same everywhere in the can).
a. In order to represent the surface area of the can as a function of r, you will first need to write a secondary equation that represents the volume of the can. Solve the volume equation for h, in terms of r. Substitute the expression for h into the surface area function. The surface area function is the primary equation to be minimized. Do we have a realistic closed interval for r?
b. Graphically investigate the absolute extrema of the function on the domain. Approximate the location of the extrema value(s).
c. Analytically solve for the absolute extrema of the function. Use calculus to find the critical number(s) of the function.
d. Do the soda cans that we see at the store use these dimensions? Why do you think this is or is not the case? Is there an assumption that we made that might not be a valid assumption?
Cleo Cola, a new Fortune 500 company, plans to sell soda pop in 12-ounce cylindrical cans. Cleo Cola will generously reward the employee team that designs the can that can be constructed with the least amount of material.
Your team is challenged to determine the dimensions that will minimize the surface area of the can. (Your team knows that 12 liquid ounces is approximately equal to 355 cubic centimeters.) Assume that the thickness of the material is uniform (i.e., the thickness of the aluminum is the same everywhere in the can).
a. In order to represent the surface area of the can as a function of r, you will first need to write a secondary equation that represents the volume of the can. Solve the volume equation for h, in terms of r. Substitute the expression for h into the surface area function. The surface area function is the primary equation to be minimized. Do we have a realistic closed interval for r?
b. Graphically investigate the absolute extrema of the function on the domain. Approximate the location of the extrema value(s).
c. Analytically solve for the absolute extrema of the function. Use calculus to find the critical number(s) of the function.
d. Do the soda cans that we see at the store use these dimensions? Why do you think this is or is not the case? Is there an assumption that we made that might not be a valid assumption?
