Final answer:
To minimize the amount of material needed to manufacture the can, you should choose the height and radius in a specific way. By using the volume formula of the cylindrical can, you can find the relationship between the height and radius. Then, you can minimize the surface area of the can by finding the critical points of an equation and using the second derivative test.
Step-by-step explanation:
To minimize the amount of material needed to manufacture the cylindrical can, both the height and radius should be chosen in a specific way. The volume of the can must be 1 liter (1000 cm³), so we can use the formula V = πr²h to find the relationship between the height and radius of the can. By rearranging the formula, we can express the height (h) in terms of the radius (r) as h = 1000 / (πr²).
To minimize the amount of material, we want to minimize the surface area of the can. The surface area of a cylindrical can is given by A = 2πrh + 2πr².
By substituting the expression for the height (h) into the surface area formula, we get A = 2πr(1000 / (πr²)) + 2πr² = 2000 / r + 2πr².
To minimize this function, we need to find its critical points. By taking the derivative of A with respect to r and setting it equal to zero, we can find the critical points. After finding the critical points, we can then determine whether they correspond to a minimum or maximum by using the second derivative test.
By following these steps, you can determine the height and radius that minimize the amount of material needed to manufacture the can.