The ratio of the curved surface area of the cylinder and the curved surface area of cone is 8: P. If slant height of the cone is 15 cm, and the total volume of the structure made by joining the same cone and cylinder such that the base of both cone and cylinder completely coincide is `1944pi cm^(3).` then find the value of P, if the radius of each of cone and cylinder is 9 cm.

Options

A. A)`5/2`
B. B)`4`
C. C)`9/2`
D. D)`3`

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas for the curved surface area and volume of the cylinder and cone.

### Step 1: Identify the given data
- Slant height of the cone (l) = 15 cm
- Radius of both cone and cylinder (r) = 9 cm
- Total volume of the structure (V_total) = 1944π cm³
- Ratio of the curved surface area of the cylinder to that of the cone = 8 : P

### Step 2: Calculate the height of the cone
To find the height of the cone (h_cone), we can use the Pythagorean theorem:
\[ h_{\text{cone}} = \sqrt{l^2 - r^2} \]
Substituting the values:
\[ h_{\text{cone}} = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12 \text{ cm} \]

### Step 3: Set up the volume equations
The total volume of the structure is the sum of the volumes of the cone and the cylinder:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{cone}} \]
Where:
- Volume of the cylinder \( V_{\text{cylinder}} = \pi r^2 h_{\text{cylinder}} \)
- Volume of the cone \( V_{\text{cone}} = \frac{1}{3} \pi r^2 h_{\text{cone}} \)

### Step 4: Substitute the known values into the volume equation
Let \( h_{\text{cylinder}} \) be the height of the cylinder. Then:
\[ 1944\pi = \pi(9^2)h_{\text{cylinder}} + \frac{1}{3}\pi(9^2)(12) \]
\[ 1944\pi = \pi(81)h_{\text{cylinder}} + \frac{1}{3}\pi(81)(12) \]
\[ 1944\pi = 81\pi h_{\text{cylinder}} + 324\pi \]
Dividing through by \( \pi \):
\[ 1944 = 81h_{\text{cylinder}} + 324 \]

### Step 5: Solve for the height of the cylinder
Rearranging the equation:
\[ 81h_{\text{cylinder}} = 1944 - 324 \]
\[ 81h_{\text{cylinder}} = 1620 \]
\[ h_{\text{cylinder}} = \frac{1620}{81} = 20 \text{ cm} \]

### Step 6: Calculate the curved surface areas
- Curved surface area of the cylinder \( A_{\text{cylinder}} = 2\pi rh_{\text{cylinder}} \)
\[ A_{\text{cylinder}} = 2\pi(9)(20) = 360\pi \text{ cm}^2 \]

- Curved surface area of the cone \( A_{\text{cone}} = \pi r l \)
\[ A_{\text{cone}} = \pi(9)(15) = 135\pi \text{ cm}^2 \]

### Step 7: Set up the ratio of the curved surface areas
The ratio of the curved surface area of the cylinder to that of the cone is given by:
\[ \frac{A_{\text{cylinder}}}{A_{\text{cone}}} = \frac{360\pi}{135\pi} = \frac{360}{135} \]
Simplifying this ratio:
\[ \frac{360}{135} = \frac{24}{9} = \frac{8}{3} \]

### Step 8: Set the ratio equal to the given ratio
We know that:
\[ \frac{A_{\text{cylinder}}}{A_{\text{cone}}} = \frac{8}{P} \]
Setting the two ratios equal:
\[ \frac{8}{3} = \frac{8}{P} \]
Cross-multiplying gives:
\[ 8P = 24 \implies P = 3 \]

### Final Answer
The value of \( P \) is **3**.