# Optimizing Metal Cost for Cylindrical Can Construction

In summary, a manufacturer is looking to construct a cylindrical can with a volume of 100 pi in^3, where the top and bottom are stronger than the sides. Using the formula pi*h*r^2=100pi, the manufacturer can find the dimensions needed to minimize the cost of the metal. With the cost of the top and bottom being 2.5 cents per square inch and the sides being 1.35 cents per square inch, the total cost can be calculated as 270pi/r + 5pi r^2. By setting the derivative of this equation to 0, the optimal value of r can be found to minimize the cost of the metal.
Hello all

For this problem:

3. A manufacturer wishes to construct a cylindrical can to hold 100 pi in^3. The top and bottom of the can are to be stronger than the sides. The tin used in making the top and bottom will cost 2.5 cents per square inch while the metal used in making the sides will cost 1.35 cents per square inch. What dimensions should be used to minimize the cost of the metal?

pi*r^2 = 100pi
Area of sides = 2pi*r*h
Area of top and bottom: 2pi*r^2

h = 100/ r^2

2pi*r(100/r^2) + 2*pi*r^2

Now do I multiply the cost of the sides by the coefficients?

Any help is appreciated

Thanks!
courtrigrad is online now Edit/Delete Message

You have "pi r^2= 100 pi" when, of course, it should be pi h r^2= 100 pi. (You knew that since you then have h= 100/r^2).

Yes, the 'cost' is the area of each part times the cost of each part: The cost of the side will be 1.35(2pi r)(100/r^2) (which is 270pi/r) and the cost of the top and bottom will be (2.5)(2 pi r^2)= 5pi r^2. The cost of the entire can is the sum of those:
270pi/r+ 5pi r^2. Set the derivative of that equal to 0 to find the value of r that makes that a minimum. Don't forget to find h too!

thanks a lot

## What is verification of method?

Verification of method is the process of evaluating and validating a scientific or analytical method to ensure its accuracy, precision, and reliability. This is typically done before the method is used for any experiments or data analysis.

## Why is verification of method important?

Verification of method is important because it ensures the validity and reliability of the results obtained from the method. It also helps to identify any potential sources of error and allows for adjustments to be made to improve the method.

## What are the steps involved in verification of method?

The steps involved in verification of method include defining the purpose and scope of the method, establishing the performance criteria, conducting experiments to test the method, analyzing the results, and documenting the findings.

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