A cone of same height and same base radius is cut from a cylinder of height 8 cm and base radius 6 cm .Find the total surface area and volume of the remaining solid.

Answer

To solve the problem of finding the total surface area and volume of the remaining solid after cutting a cone from a cylinder, we will follow these steps:

### Step 1: Find the Volume of the Cylinder
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Height \( h = 8 \) cm
- Radius \( r = 6 \) cm

Substituting the values:
\[ V_{\text{cylinder}} = \pi (6)^2 (8) = \pi (36)(8) = 288\pi \, \text{cm}^3 \]

### Step 2: Find the Volume of the Cone
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

Since the cone has the same height and radius as the cylinder:
- Height \( h = 8 \) cm
- Radius \( r = 6 \) cm

Substituting the values:
\[ V_{\text{cone}} = \frac{1}{3} \pi (6)^2 (8) = \frac{1}{3} \pi (36)(8) = \frac{288}{3}\pi = 96\pi \, \text{cm}^3 \]

### Step 3: Find the Volume of the Remaining Solid
To find the volume of the remaining solid, we subtract the volume of the cone from the volume of the cylinder:
\[ V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = 288\pi - 96\pi = 192\pi \, \text{cm}^3 \]

### Step 4: Calculate the Total Surface Area of the Remaining Solid
The total surface area \( A \) of the remaining solid includes:
1. The area of the base of the cylinder (which remains).
2. The lateral surface area of the cylinder.
3. The lateral surface area of the cone.

#### Area of the Base of the Cylinder
\[ A_{\text{base}} = \pi r^2 = \pi (6)^2 = 36\pi \, \text{cm}^2 \]

#### Lateral Surface Area of the Cylinder
\[ A_{\text{cylinder}} = 2\pi rh = 2\pi (6)(8) = 96\pi \, \text{cm}^2 \]

#### Lateral Surface Area of the Cone
To find the lateral surface area of the cone, we first need the slant height \( l \):
\[ l = \sqrt{r^2 + h^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \, \text{cm} \]
Then, the lateral surface area of the cone is:
\[ A_{\text{cone}} = \pi r l = \pi (6)(10) = 60\pi \, \text{cm}^2 \]

### Step 5: Total Surface Area of the Remaining Solid
Now, we can sum these areas to find the total surface area:
\[ A_{\text{total}} = A_{\text{base}} + A_{\text{cylinder}} + A_{\text{cone}} = 36\pi + 96\pi + 60\pi = 192\pi \, \text{cm}^2 \]

### Final Answers
1. Volume of the remaining solid: \( 192\pi \, \text{cm}^3 \) or approximately \( 602.88 \, \text{cm}^3 \) (using \( \pi \approx 3.14 \)).
2. Total surface area of the remaining solid: \( 192\pi \, \text{cm}^2 \) or approximately \( 602.88 \, \text{cm}^2 \).