Answer:
160 square inches
Step-by-step explanation:
To find the minimum surface area of the container, we need to minimize the surface area function subject to the volume constraint.
Let's denote:
- The radius of the open top of the cylinder as 'r'
- The height of the cylinder as 'h'
The surface area (A) of the open top cylinder is given by:
A = 2πr^2 + 2πrh
We need to find the minimum value of A such that the volume V = πr^2h equals 64π cubic inches.
To minimize A, we can express A in terms of a single variable, for example, in terms of 'r' or 'h', using the volume constraint equation.
Solving for the height in terms of the radius from the volume equation:
h = 64 / (πr^2)
Now, we substitute this value of 'h' into the equation for 'A' to express the surface area as a function of one variable, 'r':
A(r) = 2πr^2 + 2πr(64 / (πr^2))
A(r) = 2πr^2 + 128 / r
To find the minimum surface area, we take the derivative of A with respect to 'r', set it to zero, and solve for 'r'.
A'(r) = 4πr - 128 / r^2
0 = 4πr - 128 / r^2
4πr = 128 / r^2
r^3 = 32 / π
r = (32 / π)^(1/3)
To ensure this is a minimum, we can use the second derivative test. After verifying that it's a minimum, we can find the corresponding height using the volume equation.
h = 64 / (π(32 / π)^(2/3))
h = 8 / (32 / π)^(1/3)
h = 8 (π / 32)^(1/3)
Finally, we can substitute the values of 'r' and 'h' back into the surface area equation to find the minimum surface area.
A = 2π((32 / π)^(1/3))^2 + 2π((32 / π)^(1/3))(8 (π / 32)^(1/3))
A = 2π(32 / π)^(2/3) + 16π(32 / π)^(1/3)
A ≈ 160 square inches
