The radius of the base and the height of a right circular cone are 7 cm and 24 cm respectively. Find the volume and the total surface area of the cone.

Answer

To find the volume and total surface area of a right circular cone with a radius of 7 cm and a height of 24 cm, we can follow these steps:

### Step 1: Calculate the Volume of the Cone
The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone
- \( h \) is the height of the cone

Substituting the values:

\[ V = \frac{1}{3} \times \frac{22}{7} \times (7)^2 \times (24) \]

### Step 2: Simplify the Volume Calculation
Calculating \( (7)^2 \):

\[ (7)^2 = 49 \]

Now substituting back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 49 \times 24 \]

### Step 3: Cancel and Calculate
We can cancel \( 49 \) with \( 7 \):

\[ 49 \div 7 = 7 \]

So now we have:

\[ V = \frac{1}{3} \times 22 \times 7 \times 24 \]

Next, simplify \( 24 \div 3 \):

\[ 24 \div 3 = 8 \]

Now substituting:

\[ V = 22 \times 7 \times 8 \]

### Step 4: Final Calculation for Volume
Calculating \( 22 \times 7 \):

\[ 22 \times 7 = 154 \]

Now multiply by \( 8 \):

\[ 154 \times 8 = 1232 \]

Thus, the volume of the cone is:

\[ V = 1232 \, \text{cm}^3 \]

### Step 5: Calculate the Slant Height
The slant height \( l \) of the cone can be calculated using the Pythagorean theorem:

\[ l = \sqrt{h^2 + r^2} \]

Substituting the values:

\[ l = \sqrt{(24)^2 + (7)^2} \]

Calculating \( (24)^2 \) and \( (7)^2 \):

\[ (24)^2 = 576, \quad (7)^2 = 49 \]

Now substituting back:

\[ l = \sqrt{576 + 49} = \sqrt{625} \]

Calculating the square root:

\[ l = 25 \, \text{cm} \]

### Step 6: Calculate the Total Surface Area
The formula for the total surface area \( A \) of a cone is:

\[ A = \pi r l + \pi r^2 \]

Substituting the values:

\[ A = \frac{22}{7} \times 7 \times 25 + \frac{22}{7} \times (7)^2 \]

### Step 7: Simplify the Surface Area Calculation
First term:

\[ A = \frac{22}{7} \times 7 \times 25 = 22 \times 25 = 550 \]

Second term:

\[ A = \frac{22}{7} \times 49 = 22 \times 7 = 154 \]

### Step 8: Final Calculation for Total Surface Area
Now combine both terms:

\[ A = 550 + 154 = 704 \, \text{cm}^2 \]

### Final Answers
- Volume of the cone: \( 1232 \, \text{cm}^3 \)
- Total surface area of the cone: \( 704 \, \text{cm}^2 \)