The total surface area of a rigth circular cylinder is `165 pi cm^(2)`. If the radius of its base is 5 cm, find its heigth and volume.

Answer

To solve the problem step by step, we will follow the mathematical approach to find the height and volume of the cylinder.

### Step 1: Write down the formula for the total surface area of a cylinder.
The total surface area (TSA) of a right circular cylinder is given by the formula:
\[ \text{TSA} = 2\pi r (r + h) \]
where \( r \) is the radius and \( h \) is the height.

### Step 2: Substitute the given values into the formula.
We know that the total surface area is \( 165\pi \, \text{cm}^2 \) and the radius \( r = 5 \, \text{cm} \). Substituting these values into the formula:
\[ 165\pi = 2\pi (5)(5 + h) \]

### Step 3: Simplify the equation.
We can cancel \( \pi \) from both sides:
\[ 165 = 2(5)(5 + h) \]
This simplifies to:
\[ 165 = 10(5 + h) \]

### Step 4: Divide both sides by 10.
\[ \frac{165}{10} = 5 + h \]
This gives:
\[ 16.5 = 5 + h \]

### Step 5: Solve for \( h \).
To find \( h \), subtract 5 from both sides:
\[ h = 16.5 - 5 \]
\[ h = 11.5 \, \text{cm} \]

### Step 6: Calculate the volume of the cylinder.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \pi (5^2)(11.5) \]
Calculating \( 5^2 \):
\[ V = \pi (25)(11.5) \]

### Step 7: Multiply to find the volume.
Calculating \( 25 \times 11.5 \):
\[ 25 \times 11.5 = 287.5 \]
Thus, the volume is:
\[ V = 287.5\pi \, \text{cm}^3 \]

### Final Answer:
- Height \( h = 11.5 \, \text{cm} \)
- Volume \( V = 287.5\pi \, \text{cm}^3 \)