Final answer:
To find the volume of the cylinder with a given surface area, we can use the formula V = πr²h. By substituting the given values and solving for the unknowns, we can calculate the volume of the cylinder.
Step-by-step explanation:
To find the volume of the cylinder, we can use the formula V = πr²h, where r is the radius and h is the height of the cylinder. We are given that the curved surface area is one third of the total surface area, so we can determine the curved surface area as follows:
Curved Surface Area = Total Surface Area - 2πr²
Since the curved surface area is one third of the total surface area, we have:
Curved Surface Area = (1/3) * Total Surface Area
Substituting the formulas for curved surface area and total surface area, we get:
(1/3) * Total Surface Area = Total Surface Area - 2πr²
Simplifying the equation gives:
2πr² = (2/3) * Total Surface Area
Now, we can substitute the given value of the total surface area (462 cm²) into the equation and solve for r:
2πr² = (2/3) * 462
r² = (1/3) * 462 / π
r ≈ √(154 / π)
Once we have the value of r, we can substitute it into the formula for volume and solve for V:
V = πr²h
Substituting the given values of r and h (5.25 cm), we get:
V ≈ π(√(154 / π))² * 5.25
Using the approximation of π as 3.142, we can calculate the volume of the cylinder to be approximately 26.736 cm³.