Final answer:
The radius of the cylindrical can is approximately 2.9 cm and the height is approximately 5.22 cm.
Step-by-step explanation:
To find the radius and height of a cylindrical can with the largest possible volume, we need to consider the total surface area and the formula for volume of a cylinder.
The formula for the total surface area of a cylinder is A = 2πrh + 2πr², where A is the total surface area, r is the radius, and h is the height of the cylinder.
In this case, we are given that the total surface area is 160 cm². We can equate this to the formula and solve for h:
160 = 2πrh + 2πr²
From here, we can either solve for r in terms of h or h in terms of r. Let's solve for h in terms of r:
- 160 = 2πrh + 2πr²
- 160 - 2πr² = 2πrh
- h = (160 - 2πr²) / (2πr)
Now, we can substitute the value of h into the volume formula and simplify to find the derivative:
- V = πr²h
- V = πr²((160 - 2πr²) / (2πr))
- V = (160r - 2πr³) / 2
To find the maximum volume, we need to find where the derivative equals zero. This occurs when:
- 0 = 160 - 6πr²
- r² = 160 / (6π)
- r² ≈ 8.49
- r ≈ √8.49
- r ≈ 2.9
Now that we have the value of r, we can substitute it back into the equation for h:
- h = (160 - 2π(2.9)²) / (2π(2.9))
- h ≈ 5.22
Therefore, the radius of the cylindrical can is approximately 2.9 cm and the height is approximately 5.22 cm.