Curved surface area of Cone P is seven times the curved surface area of Cone Q. Slant height of Cone Q is seven times the slant height of Cone P. What will be the ration of the area base of Cone P to the area of the base of Cone Q?

Options

A. `2401 : 1`
B. `343 : 1`
C. `49 : 1`
D. `7 : 1`

Answer

To solve the problem step by step, we need to analyze the given information about the two cones, P and Q.

### Step 1: Understand the given information
We know that:
- The curved surface area (CSA) of Cone P is 7 times that of Cone Q.
- The slant height (l) of Cone Q is 7 times that of Cone P.

Let:
- CSA of Cone Q = \( S_Q \)
- CSA of Cone P = \( S_P = 7 \times S_Q \)

### Step 2: Write the formula for the curved surface area of a cone
The formula for the curved surface area of a cone is given by:
\[ S = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 3: Set up the equations based on the given information
From the information provided:
1. For Cone Q:
\[ S_Q = \pi r_Q l_Q \]
2. For Cone P:
\[ S_P = \pi r_P l_P = 7 \times S_Q \]

### Step 4: Substitute the slant height relationship
Given that \( l_Q = 7 \times l_P \), we can substitute this into the equation for \( S_Q \):
\[ S_Q = \pi r_Q (7 l_P) = 7 \pi r_Q l_P \]

### Step 5: Substitute \( S_Q \) into the equation for \( S_P \)
Now substituting \( S_Q \) into the equation for \( S_P \):
\[ S_P = 7 \times S_Q = 7 \times (7 \pi r_Q l_P) = 49 \pi r_Q l_P \]
Now we can express \( S_P \) in terms of \( r_P \) and \( l_P \):
\[ S_P = \pi r_P l_P \]

### Step 6: Equate the two expressions for \( S_P \)
Setting the two expressions for \( S_P \) equal to each other:
\[ \pi r_P l_P = 49 \pi r_Q l_P \]
Dividing both sides by \( \pi l_P \) (assuming \( l_P \neq 0 \)):
\[ r_P = 49 r_Q \]

### Step 7: Find the ratio of the areas of the bases
The area of the base of a cone is given by:
\[ A = \pi r^2 \]
Thus, the area of the base of Cone P is:
\[ A_P = \pi r_P^2 = \pi (49 r_Q)^2 = \pi \times 2401 r_Q^2 \]
And the area of the base of Cone Q is:
\[ A_Q = \pi r_Q^2 \]

### Step 8: Set up the ratio of the areas of the bases
Now, we can find the ratio of the area of the base of Cone P to that of Cone Q:
\[ \text{Ratio} = \frac{A_P}{A_Q} = \frac{2401 \pi r_Q^2}{\pi r_Q^2} = 2401 \]

### Final Answer
Thus, the ratio of the area of the base of Cone P to the area of the base of Cone Q is:
\[ \text{Ratio} = 2401 : 1 \]