A garbage can is in the shape of a cylinder with no lid. It needs to have a volume of 5000 cm. What will be the radius and height of the can that uses 3 the least amount of material to construct it? *(using surface area)

To minimize the material needed to construct the garbage can, we need to minimize its surface area. Since the can has no lid, its surface area is given by: A = 2πrh + 2πr^2 where r is the radius of the base of the cylinder, h is the height, and π is a constant (approximately 3.14). We are given that the volume of the can should be 5000 cm^3, so we can use the formula for the volume of a cylinder: V = πr^2h to write h in terms of r: h = V/πr^2 Substituting this expression for h into the equation for the surface area, we get: A = 2πr(V/πr^2) + 2πr^2 A = 2V/r + 2πr^2 To minimize A, we take its derivative with respect to r and set it equal to zero: dA/dr = -2V/r^2 + 4πr = 0 2V = r^3π Solving for r, we get: r = (2V/π)^(1/3) r = (2*5000/π)^(1/3) ≈ 12.3 cm Substituting this value of r into the expression for h in terms of r, we get: h = V/πr^2 h = 5000/(π*(12.3)^2) ≈ 10.2 cm Therefore, the garbage can with a volume of 5000 cm^3 that uses the least amount of material to construct it has a radius of approximately 12.3 cm and a height of approximately 10.2 cm.