Final answer:
The amount of material required to make the drum can be expressed as the surface area of the cylinder. The surface area formula is A = 200/r + 2πr².
Step-by-step explanation:
(a) The amount of material required to make the drum can be expressed as the surface area of the cylinder. The surface area of a right circular cylinder is given by the formula A = 2πrh + 2πr², where r is the radius and h is the height. Since the drum is required to have a volume of 100 cubic feet, we can use the formula for the volume of a cylinder, V = πr²h, to find that h = 100/(πr²). Substituting this value of h into the surface area formula gives A = 2πr(100/(πr²)) + 2πr² = 200/r + 2πr².
(b) If the drum's radius is 3 feet, we can substitute r = 3 into the surface area formula A = 200/r + 2πr² to find the amount of material required.
(c) If the drum's radius is 4 feet, we can substitute r = 4 into the surface area formula A = 200/r + 2πr² to find the amount of material required.
(d) If the drum's radius is 5 feet, we can substitute r = 5 into the surface area formula A = 200/r + 2πr² to find the amount of material required.
(e) To find the value of r that results in the smallest amount of material required, we can take the derivative of the surface area formula A = 200/r + 2πr² with respect to r, set it equal to zero, and solve for r.