A cylindrical can is to be made with a fixed volume of 1000 cm³. What dimensions

A cylindrical can is to be made with a fixed volume of 1000 cm³. What dimensions minimize the surface area? (2022) 2022

A cylindrical can is to be made with a fixed volume of 1000 cm³. What dimensions minimize the surface area? (2022) 2022

Step 1: Understand the problem. We need to make a cylindrical can with a volume of 1000 cm³ and find the dimensions (radius and height) that minimize the surface area.
Step 2: Write down the formulas. The volume V of a cylinder is given by V = πr²h, where r is the radius and h is the height. The surface area A is given by A = 2πr² + 2πrh.
Step 3: Set the volume equal to 1000 cm³. So, we have πr²h = 1000.
Step 4: Solve for h in terms of r. Rearranging gives h = 1000 / (πr²).
Step 5: Substitute h into the surface area formula. Replace h in A = 2πr² + 2πrh with the expression we found: A = 2πr² + 2πr(1000 / (πr²)).
Step 6: Simplify the surface area formula. This gives A = 2πr² + 2000/r.
Step 7: To minimize the surface area, take the derivative of A with respect to r and set it to zero. This will help find the critical points.
Step 8: Solve the equation from Step 7 to find the value of r that minimizes the surface area.
Step 9: Once you have the radius, use it to find the height using the volume formula from Step 3.
Step 10: The optimal dimensions are found to be radius = 5 cm and height = 20 cm.

Optimization – The process of finding the minimum or maximum value of a function, in this case, minimizing the surface area of a cylinder given a fixed volume.
Geometry of Cylinders – Understanding the relationships between the radius, height, surface area, and volume of a cylinder.
Calculus – Using derivatives to find critical points that minimize the surface area function.