Final answer:
To find the radius and height of the cylindrical can with the maximum volume and a total surface area of 130 cm², we can use optimization techniques. By expressing the height in terms of the radius using the surface area formula, we can find the maximum volume by finding the derivative of the volume equation. The calculated radius and height will give us the dimensions of the can with the largest possible volume.
Step-by-step explanation:
To find the radius and height of the cylindrical can with the maximum volume and a total surface area of 130 cm², we need to use optimization techniques. The volume of a cylindrical can is given by V = πr²h, where r is the radius and h is the height.
To maximize the volume, we can use the surface area formula to express the height h in terms of the radius r. The total surface area of a cylinder is given by A = 2πrh + 2πr². Rearranging this equation to solve for h, we get h = (A - 2πr²) / 2πr.
Substitute the value of h in terms of r into the volume equation, and we get V = πr²((A - 2πr²) / 2πr). Simplify the expression and find the derivative of V with respect to r. Set the derivative equal to zero and solve for r to find the radius. Once the radius is known, substitute it back into the equation for h to find the height.