Final answer:
To find the dimensions of a cylindrical can with a volume of 500 in³ that minimizes the surface area, we can use the formula for surface area of a cylinder and the constraint equation for volume. By taking the derivative of the surface area with respect to the radius, setting it equal to 0, and solving for the radius, we can find the value of the radius that minimizes the surface area. The corresponding height can be found by substituting this value of radius back into the constraint equation.
Step-by-step explanation:
To find the dimensions of a cylindrical can with a volume of 500 in³ that minimizes the surface area, we first need to calculate the dimensions that will give us the minimum surface area. The surface area of a cylinder is given by the formula S = 2πr² + 2πrh.
Let's assume the radius of the cylinder is r and the height is h. We need to minimize S subject to the constraint V = πr²h = 500 in³.
Using the constraint equation, we can solve for either the radius or the height in terms of the other variable, and substitute it into the formula for surface area. Let's solve for h in terms of r:
h = 500 / (πr²)
Substituting this expression for h into the formula for surface area:
S = 2πr² + 2πr(500 / (πr²))
Simplifying the equation:
S = 2πr² + 1000 / r
To find the minimum value of S, we can take the derivative of S with respect to r, set it equal to 0, and solve for r:
dS/dr = 4πr - 1000 / r² = 0
Simplifying the equation:
4πr³ - 1000 = 0
Dividing both sides by 4π:
r³ = 250 / π
Taking the cube root of both sides:
r = (250 / π)^(1/3)
So the radius that will minimize the surface area is approximately (250 / π)^(1/3). To find the corresponding height, we can substitute this value of r back into the constraint equation:
h = 500 / (π(250 / π)^(2/3)) = 2 * (250 / π)^(2/3).