Total surface area of a cylinder mounted with a hemispherical bowl on one end is 2552 cm2. If height of cylinder is 8 cm then find the volume of the solid body?(`cm^(3)`)

Options

A. `10443 (1)/(3)`
B. `10677 (1)/(3)`
C. `10547 (1)/(3)`
D. `10977(1)/(3)`

Answer

To solve the problem step by step, we need to find the volume of a solid body that consists of a cylinder with a hemispherical bowl on one end, given the total surface area and the height of the cylinder.

### Step 1: Understand the given information
- Total Surface Area (TSA) of the solid = 2552 cm²
- Height of the cylinder (h) = 8 cm
- We need to find the radius (r) of the cylinder and the hemispherical bowl.

### Step 2: Write the formula for the Total Surface Area
The total surface area of the cylinder with a hemispherical bowl on one end is given by:
\[ \text{TSA} = 2\pi r^2 + 2\pi rh + \pi r^2 \]
Where:
- \(2\pi r^2\) is the curved surface area of the cylinder,
- \(2\pi rh\) is the lateral surface area of the cylinder,
- \(\pi r^2\) is the area of the base of the hemisphere.

Combining the terms, we have:
\[ \text{TSA} = 3\pi r^2 + 2\pi rh \]

### Step 3: Substitute the known values into the TSA equation
Substituting the known values into the TSA equation:
\[ 3\pi r^2 + 2\pi r(8) = 2552 \]
This simplifies to:
\[ 3\pi r^2 + 16\pi r = 2552 \]

### Step 4: Factor out \(\pi\)
Dividing the entire equation by \(\pi\) (using \(\pi \approx 22/7\) or \(3.14\)):
\[ 3r^2 + 16r = \frac{2552}{\pi} \]
Using \(\pi \approx 22/7\):
\[ \frac{2552 \times 7}{22} = 812 \]
So we have:
\[ 3r^2 + 16r - 812 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 3\), \(b = 16\), and \(c = -812\):
\[ b^2 - 4ac = 16^2 - 4 \times 3 \times (-812) \]
Calculating:
\[ = 256 + 97224 = 97480 \]
Now applying the quadratic formula:
\[ r = \frac{-16 \pm \sqrt{97480}}{6} \]
Calculating \(\sqrt{97480} \approx 312.2\):
\[ r = \frac{-16 \pm 312.2}{6} \]
Calculating the two possible values for \(r\):
1. \(r = \frac{296.2}{6} \approx 49.37\) (valid)
2. \(r = \frac{-328.2}{6}\) (not valid as radius cannot be negative)

### Step 6: Calculate the volume of the solid
The volume \(V\) of the solid body is given by the sum of the volume of the cylinder and the volume of the hemisphere:
\[ V = \text{Volume of Cylinder} + \text{Volume of Hemisphere} \]
\[ V = \pi r^2 h + \frac{2}{3}\pi r^3 \]
Substituting \(r = 14\) and \(h = 8\):
\[ V = \pi (14^2)(8) + \frac{2}{3}\pi (14^3) \]
Calculating:
\[ = \pi (196)(8) + \frac{2}{3}\pi (2744) \]
\[ = 1568\pi + \frac{5488}{3}\pi \]
\[ = \frac{4704 + 5488}{3}\pi = \frac{10192}{3}\pi \]

### Final Volume Calculation
Using \(\pi \approx 3.14\):
\[ V \approx \frac{10192}{3} \times 3.14 \approx 10677.33 \text{ cm}^3 \]

### Conclusion
The volume of the solid body is approximately \(10677.33 \text{ cm}^3\).