The total surface area of a right circular cylinder is 165π cm². If the radius of its base is 5 cm, find its height and volume.

We are given that the total surface area of a right circular cylinder is 165π cm², and the radius of the base is 5 cm. We need to find the height and volume of the cylinder.

Step 1: Formula for the total surface area of a cylinder
The total surface area (TSA) of a cylinder is given by the formula:

TSA = 2πr² + 2πrh

Where:

r is the radius of the base
h is the height of the cylinder
We are given that TSA = 165π cm² and r = 5 cm.

Step 2: Substitute the given values into the TSA formula
Substituting the values into the formula:

165π = 2π(5)² + 2π(5)(h)

Simplify the terms:

165π = 2π(25) + 2π(5h)

165π = 50π + 10πh

Step 3: Solve for the height (h)
Now, subtract 50π from both sides:

165π - 50π = 10πh

115π = 10πh

Cancel the π on both sides:

115 = 10h

Now, solve for h:

h = 115 / 10

h = 11.5 cm

So, the height of the cylinder is 11.5 cm.

Step 4: Formula for the volume of a cylinder
The volume (V) of a cylinder is given by the formula:

V = πr²h

Substitute the values of r = 5 cm and h = 11.5 cm:

V = π(5)²(11.5)

V = π(25)(11.5)

V = 287.5π cm³

Thus, the volume of the cylinder is 287.5π cm³.

Final Answer:
The height of the cylinder is 11.5 cm.
The volume of the cylinder is 287.5π cm³.