Final answer:
To find the radius and height of the box with minimum surface area, we first find the volume using the formula V = πr²h. Then, we find the surface area using the formula A = 2πrh + πr².
Taking the derivative of the surface area function and setting it equal to zero, we can solve for the radius. Substituting the radius back into the equation for the height gives us the answer.
Step-by-step explanation:
To find the radius of the base and the height of the box with minimum surface area, we first find the volume of the box. The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius of the base, and h is the height. In this case, the volume is given as 8π. So we have 8π = πr²h. Solving for h, we get h = 8/r².
Next, we need to find the surface area of the box. The formula for the surface area of a cylinder is A = 2πrh + πr², where A is the surface area. Substituting the value of h we found, we have A = 2πr(8/r²) + πr² = 16π/r + πr².
To find the minimum surface area, we need to find the derivative of the surface area function and set it equal to zero. Differentiating A with respect to r, we get dA/dr = -16π/r² + 2πr. Setting this equal to zero and solving for r, we find r = 4/√π. Substituting this value of r back into the equation for h, we get h = 8/(4/√π)² = 2π.
Therefore, the radius of the base is 4/√π and the height of the box with minimum surface area is 2π.