Real World Applications
Minimize Cost of a Can/Minimize Acid in a Reservoir
Finding the Minimal Cost of a Coffee Can
Grinds Coffee company manufactures aluminum cans in the shape of a cylinder with a maximum capacity of 135 cubic inches. The top and bottom of the coffee can uses a special aluminum alloy that costs 6 cents per square inch. The sides of the coffee can are made with cheaper aluminum that costs 4 cents per square inch.
- Create a function that represents the cost of the can as a function of the radius of the can.
- Use your TI-83 to graph the function F(r)
- What value of r will result in the least cost?
- What is the least cost?
Solution
Observe the picture of a coffee can broken up into its parts along with the respective formulas:
 |
©Prentice Hall --Precalculus (Graphing and Data Analysis) |
This problem has one more requirement. The radius and height must be chosen so that the volume of the can is 135 cubic inches. This results in the following set of calculations and the following TI-83 graphs (the second graph shows the minimum):
 |
 |
 |
 |
The cost for the coffee cans is least for a radius of about 2.4 inches. The least cost is about 6.7 cents!
You try
A diet company packages powder for a milkshake product. The lids are made of a cardboard product and cost 2.5 cents per square centimeter. The sides of the container are made of a slightly more expensive plastic product. The material for the sides costs 4 cents per square centimeter. The containers must hold 1000 cubic centimeters of powder.
- Create a function that represents the cost of the can as a function of the radius of the can.
- Use your TI-83 to graph the function C(r)
- What value of r will result in the least cost?
- What is the least cost?
- What are the dimensions for the can with the least cost?
- Do your least cost dimensions make a practical can? Draw a picture to support your conclusion.
- Can you find any cans that use more expensive material for the lateral surface and cheaper material for the lids? How could you find this information?
- Can you find any cans that use cheaper material for the lateral surface and more expensive material for the lids? How could you find this information?
Minimize Acid in a Reservoir
A reservoir contains 150,000 gallons of water with an acidity of 5.5%. Find a formula for acidity and complete the table below:
| Added Water | 100,000 | 250,000 | 300,000 | 400,000 |
| Acidity | ? | ? | ? | ? |
| Amount | % Acid | Amount of Acid | |
| Original | 150,000 | .055 | 8250 |
| Added Water | x | 0 | 0 |
| New Solution | 150,000+x | y | y(150,000+x) |
The amount of the Acid does not change in the reservoir.
Since we are adding water to the original solution we can say:
8250=y(150,000+x) or y=8250/(150,000+x)
After entering the function into our calculator we can use tables to chart data as follows:
Complete the acidity chart! Suppose the goal is to reduce the acid level in the reservoir to very tiny levels. Is it possible to eliminate the acid by adding enough water? Why or why not? What is the meaning of the asymptote in this rational equation?
You Try:
Water is mixed with 35 milliliters of a 45% acid solution. Write the acid concentration as a function of the amount of water added. Graph this function on your TI-83 calculator. What will be the concentration when 20 milliliters have been added to the solution?