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)
Accepted Solution
A:
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.