Final answer:
the radius is approximately 1.361 cm.
Step-by-step explanation:
To find the radius of the cylinder that produces the minimum surface area, we need to minimize the surface area given the volume of 6 cubic centimeters.
Let's denote the radius of the cylinder as 'r' and the height as 'h'. We can write the volume of the cylinder as V = πr²h. We can also express the surface area of the cylinder as A = 2πr² + 2πrh.
Since we want to minimize the surface area, we need to minimize the function A = 2πr² + 2πrh while keeping the volume V = 6 constant.
To proceed further, we can use the volume formula to express the height in terms of the radius, h = (6 / (πr²)). Substituting this into the surface area equation, we get A = 2πr² + 2πr(6 / (πr²)).
Simplifying further, A = 2πr² + 12/r. To find the minimum surface area, we can take the derivative of A with respect to r, set it equal to zero, and solve for r.
dA/dr = 4πr - 12/r² = 0
4πr = 12/r²
4πr³ = 12
r³ = 3 / π
r = (3 / π)^(1/3)
Substituting the value of π as approximately 3.142, we can calculate the value of r as approximately 1.361.