From a circular cylinder of diameter 10 cm and height 12 cm a conical cavity of the same base radius and of the same height in hollowed out. Find the volume and the whole surfce of the remianing solid .Leave the answer in `pi`.

Answer

To solve the problem, we need to find the volume and the total surface area of the remaining solid after a conical cavity is hollowed out from a circular cylinder. Let's break this down step by step.

### Step 1: Identify the dimensions of the cylinder and the cone
- **Diameter of the cylinder** = 10 cm
- **Radius of the cylinder (r)** = Diameter / 2 = 10 cm / 2 = 5 cm
- **Height of the cylinder (h)** = 12 cm
- The cone that is hollowed out has the same base radius and height as the cylinder.

### Step 2: Calculate the volume of the cylinder
The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
Substituting the values:
\[ V_{\text{cylinder}} = \pi (5^2)(12) = \pi (25)(12) = 300\pi \, \text{cm}^3 \]

### Step 3: Calculate the volume of the cone
The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V_{\text{cone}} = \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = 100\pi \, \text{cm}^3 \]

### Step 4: Calculate the volume of the remaining solid
The volume of the remaining solid is given by:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} \]
Substituting the volumes:
\[ V_{\text{remaining}} = 300\pi - 100\pi = 200\pi \, \text{cm}^3 \]

### Step 5: Calculate the total surface area of the remaining solid
The total surface area (TSA) of the remaining solid can be calculated as follows:
1. The TSA of the cylinder is given by:
\[ \text{TSA}_{\text{cylinder}} = 2\pi r (r + h) \]
2. The area of the base of the cylinder (which is removed) is:
\[ \text{Area}_{\text{base}} = \pi r^2 \]
3. The curved surface area (CSA) of the cone is:
\[ \text{CSA}_{\text{cone}} = \pi r l \]
where \( l \) is the slant height of the cone, calculated using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{cm} \]

### Step 6: Substitute values to find TSA of the remaining solid
Now we can substitute the values:
1. TSA of the cylinder:
\[ \text{TSA}_{\text{cylinder}} = 2\pi (5)(5 + 12) = 2\pi (5)(17) = 170\pi \, \text{cm}^2 \]
2. Area of the base removed:
\[ \text{Area}_{\text{base}} = \pi (5^2) = 25\pi \, \text{cm}^2 \]
3. CSA of the cone:
\[ \text{CSA}_{\text{cone}} = \pi (5)(13) = 65\pi \, \text{cm}^2 \]

### Final Calculation of TSA
The TSA of the remaining solid is:
\[ \text{TSA}_{\text{remaining}} = \text{TSA}_{\text{cylinder}} - \text{Area}_{\text{base}} + \text{CSA}_{\text{cone}} \]
Substituting the values:
\[ \text{TSA}_{\text{remaining}} = 170\pi - 25\pi + 65\pi = 210\pi \, \text{cm}^2 \]

### Conclusion
- **Volume of the remaining solid** = \( 200\pi \, \text{cm}^3 \)
- **Total surface area of the remaining solid** = \( 210\pi \, \text{cm}^2 \)