Final answer:
The radius that minimizes the surface area of a 250 cm³ cylindrical can is determined using calculus by finding where the derivative of the surface area formula, in terms of the radius, equals zero.
Step-by-step explanation:
A student wants to know the radius that will minimize the surface area of a cylindrical can designed to contain 250 cm³ of liquid. To solve this, we can use calculus and the formulas for the volume and surface area of a cylinder. The volume (V) of a cylinder with radius (r) and height (h) is given by V = πr²h, and the total surface area (A) which includes the base and the side, is given by A = 2πr² + 2πrh. Given that V = 250 cm³, we can solve for h in terms of r, yielding h = 250 / (πr²). Substituting this expression for h back into the surface area formula, we get A = 2πr² + 2πr(250 / (πr²)) = 2πr² + 500/r. To minimize the surface area, we can take the derivative of A with respect to r, set it equal to zero, and solve for r. After finding the critical points, we can test the values to determine which gives the minimum surface area.