A cylindrical can (w/lid) needs to be construced to hold 40 cbic inches. Find the dimentions of the can so that the total amout of material needed is a minimum
i don't know how to do this!!
please help- do you know any sites that reteach this?
To find the dimensions of the cylindrical can that minimize the total amount of material needed, we can use calculus. Specifically, we need to find the critical points of the function that represents the total amount of material used.
To begin, let's define the variables:
- Let r be the radius of the circular base of the can.
- Let h be the height of the can.
- Let V be the volume of the can, which is given as 40 cubic inches.
- Let A be the surface area of the can, which represents the amount of material used.
First, we need to express the volume of the can in terms of r and h. Since the can is cylindrical, the volume can be calculated using the formula V = πr^2h.
Next, we need to express the surface area of the can in terms of r and h. The surface area is the sum of the areas of the circular base and the lateral surface area of the can. The base area is given by πr^2, and the lateral surface area is given by 2πrh.
Therefore, the surface area A can be calculated as A = πr^2 + 2πrh.
To minimize the amount of material needed, we want to minimize the surface area A while keeping the volume V constant at 40 cubic inches. This can be achieved by finding the critical points of the function A with respect to the variables r and h.
To learn more about the calculus concepts involved in finding critical points and minimizing a function, you can refer to the following websites:
1. Khan Academy: https://www.khanacademy.org/math/differential-calculus/dc-derivatives-opt-applications/dc-differentiation-applications/a/optimizing-box-volume
2. Paul's Notes: http://tutorial.math.lamar.edu/Classes/CalcI/MaxMinProblems.aspx
3. Math Is Fun: https://www.mathsisfun.com/calculus/maxima-minima.html
These websites provide explanations, examples, and practice problems to help you understand the concepts and steps involved in solving optimization problems like the one you mentioned.
Once you are familiar with the concepts, you can use the derivative of the surface area function A with respect to r or h to find the critical points. By evaluating these points and considering any boundary conditions, you can determine the dimensions of the cylindrical can that will minimize the amount of material needed.