The radius of the base of a right cone is 34 cm and its height is 84 cm . Find :
(a) slant height (b) curved surface area (c) total surface area (d) volume

Answer

To solve the problem step by step, we will find the slant height, curved surface area, total surface area, and volume of the right cone with a radius of 34 cm and a height of 84 cm.

### Step 1: Find the Slant Height (L)

The formula for the slant height \( L \) of a cone is given by:

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

Where:
- \( r \) is the radius of the base of the cone (34 cm)
- \( h \) is the height of the cone (84 cm)

Substituting the values:

\[ L = \sqrt{34^2 + 84^2} \]

Calculating \( 34^2 \) and \( 84^2 \):

\[ 34^2 = 1156 \quad \text{and} \quad 84^2 = 7056 \]

Now, adding these values:

\[ L = \sqrt{1156 + 7056} = \sqrt{8212} \]

Calculating the square root:

\[ L \approx 90.62 \text{ cm} \]

Rounding to the nearest whole number:

\[ L \approx 91 \text{ cm} \]

### Step 2: Find the Curved Surface Area (CSA)

The formula for the curved surface area \( A \) of a cone is given by:

\[ A = \pi r L \]

Using \( \pi \approx \frac{22}{7} \), \( r = 34 \) cm, and \( L = 91 \) cm:

\[ A = \frac{22}{7} \times 34 \times 91 \]

Calculating \( 34 \times 91 \):

\[ 34 \times 91 = 3094 \]

Now calculating the area:

\[ A = \frac{22 \times 3094}{7} = \frac{67968}{7} \approx 9724 \text{ cm}^2 \]

### Step 3: Find the Total Surface Area (TSA)

The formula for the total surface area \( T \) of a cone is given by:

\[ T = \pi r (r + L) \]

Substituting the values:

\[ T = \frac{22}{7} \times 34 \times (34 + 91) \]

Calculating \( 34 + 91 \):

\[ 34 + 91 = 125 \]

Now substituting back into the formula:

\[ T = \frac{22}{7} \times 34 \times 125 \]

Calculating \( 34 \times 125 \):

\[ 34 \times 125 = 4250 \]

Now calculating the total surface area:

\[ T = \frac{22 \times 4250}{7} = \frac{93500}{7} \approx 13357.14 \text{ cm}^2 \]

### Step 4: Find the Volume (V)

The formula for the volume \( V \) of a cone is given by:

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

Substituting the values:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 34^2 \times 84 \]

Calculating \( 34^2 \):

\[ 34^2 = 1156 \]

Now substituting back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 1156 \times 84 \]

Calculating \( 1156 \times 84 \):

\[ 1156 \times 84 = 97104 \]

Now calculating the volume:

\[ V = \frac{1}{3} \times \frac{22 \times 97104}{7} = \frac{22 \times 97104}{21} \]

Calculating \( 22 \times 97104 \):

\[ 22 \times 97104 = 2136328 \]

Now dividing by 21:

\[ V \approx 101728 \text{ cm}^3 \]

### Summary of Results

- (a) Slant Height \( L \approx 91 \text{ cm} \)
- (b) Curved Surface Area \( A \approx 9724 \text{ cm}^2 \)
- (c) Total Surface Area \( T \approx 13357.14 \text{ cm}^2 \)
- (d) Volume \( V \approx 101728 \text{ cm}^3 \)