To find the dimensions that will require the least amount of material, we need to minimize the surface area of the cylindrical container while maintaining a volume of 1 cubic foot.
Let's assume the radius of the cylinder is r and the height is h.
The volume of a cylinder is given by the formula V = πr^2h.
We are given that the volume V is 1 cubic foot, so we have πr^2h = 1.
To find the surface area of the cylinder, we need to consider the lateral surface area (the curved surface) and the area of the base.
The lateral surface area of a cylinder is given by the formula A_lateral = 2Ï€rh.
The area of the base is given by the formula A_base = πr^2.
The total surface area of the cylinder is given by A_total = A_lateral + A_base = 2πrh + πr^2.
Now, we need to minimize the surface area A_total, which means finding the dimensions that minimize A_total.
Since we have the constraint πr^2h = 1, we can solve for h in terms of r:
h = 1 / (Ï€r^2).
Substituting this expression for h in the equation for A_total, we have:
A_total = 2πr(1 / (πr^2)) + πr^2 = 2 / r + πr^2.
To minimize A_total, we can take the derivative of A_total with respect to r and set it equal to zero:
dA_total/dr = -2/r^2 + 2Ï€r = 0.
Simplifying this equation, we get:
2Ï€r - 2/r^2 = 0.
Multiplying through by r^2, we have:
2Ï€r^3 - 2 = 0.
Now, we can solve for r:
2Ï€r^3 = 2.
Dividing both sides by 2Ï€, we get:
r^3 = 1 / π.
Taking the cube root of both sides, we have:
r = (1 / π)^(1/3).
Substituting this value of r into the equation for h, we get:
h = 1 / (Ï€r^2) = 1 / (Ï€(1/Ï€)^(2/3)) = (1/Ï€)^(1/3).
Therefore, the dimensions that will require the least amount of material are:
Radius: r = (1 / π)^(1/3).
Height: h = (1/Ï€)^(1/3).