Question

Find the radius and height of a cylindrical can of total surface area 160 cm^2 whose volume is as large as possible

Answer 1

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:

1. 160 = 2πrh + 2πr²
2. 160 - 2πr² = 2πrh
3. h = (160 - 2πr²) / (2πr)

Now, we can substitute the value of h into the volume formula and simplify to find the derivative:

1. V = πr²h
2. V = πr²((160 - 2πr²) / (2πr))
3. V = (160r - 2πr³) / 2

To find the maximum volume, we need to find where the derivative equals zero. This occurs when:

1. 0 = 160 - 6πr²
2. r² = 160 / (6π)
3. r² ≈ 8.49
4. r ≈ √8.49
5. r ≈ 2.9

Now that we have the value of r, we can substitute it back into the equation for h:

1. h = (160 - 2π(2.9)²) / (2π(2.9))
2. h ≈ 5.22

Therefore, the radius of the cylindrical can is approximately 2.9 cm and the height is approximately 5.22 cm.