The total surface area of a solid right circular cylinder is `1617cm^2 ` . If the diameter of its base is 21cm, then what is its volume (in `cm^2`)

Options

A. 5324
B. 3971
C. 4851
D. 5228

Answer

To solve the problem, we need to find the volume of a right circular cylinder given its total surface area and the diameter of its base.

### Step 1: Identify the given values
- Total Surface Area (TSA) = 1617 cm²
- Diameter of the base = 21 cm

### Step 2: Calculate the radius of the base
The radius (r) is half of the diameter.
\[ r = \frac{diameter}{2} = \frac{21 \, \text{cm}}{2} = 10.5 \, \text{cm} \]

### Step 3: Write the formula for the total surface area of a cylinder
The formula for the total surface area (TSA) of a cylinder is:
\[ TSA = 2\pi r(h + r) \]
Where:
- \( r \) = radius
- \( h \) = height

### Step 4: Substitute the known values into the TSA formula
Substituting the known values into the TSA formula:
\[ 1617 = 2\pi(10.5)(h + 10.5) \]

### Step 5: Simplify the equation
First, calculate \( 2\pi(10.5) \):
\[ 2\pi(10.5) \approx 66 \, \text{(using } \pi \approx 3.14\text{)} \]
Now, the equation becomes:
\[ 1617 = 66(h + 10.5) \]

### Step 6: Solve for \( h + 10.5 \)
Divide both sides by 66:
\[ h + 10.5 = \frac{1617}{66} \approx 24.5 \]

### Step 7: Solve for height \( h \)
Subtract 10.5 from both sides:
\[ h = 24.5 - 10.5 = 14 \, \text{cm} \]

### Step 8: Calculate the volume of the cylinder
The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
Substituting the values of \( r \) and \( h \):
\[ V = \pi (10.5)^2 (14) \]
Calculating \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]
Now substituting back into the volume formula:
\[ V = \pi (110.25)(14) \approx 3.14 \times 110.25 \times 14 \]
Calculating:
\[ V \approx 3.14 \times 1543.5 \approx 4851.09 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is approximately \( 4851 \, \text{cm}^3 \).

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