The Diameters of two cones are equal. If their slant heights are in the ratio 5:4, find the ratio o their curved surface areas

Options

A. `4:5`
B. `2:3`
C. `3:2`
D. `5:4`

Answer

To find the ratio of the curved surface areas of two cones with equal diameters and slant heights in the ratio of 5:4, we can follow these steps:

### Step 1: Understand the relationship between the cones
Since the diameters of the two cones are equal, their radii will also be equal. Let the radius of both cones be \( r \).

### Step 2: Define the slant heights
Let the slant height of cone 1 be \( l_1 \) and the slant height of cone 2 be \( l_2 \). According to the problem, the ratio of their slant heights is given as:
\[ \frac{l_1}{l_2} = \frac{5}{4} \]
From this, we can express \( l_1 \) and \( l_2 \) in terms of a common variable. Let:
\[ l_1 = 5k \quad \text{and} \quad l_2 = 4k \]
for some positive constant \( k \).

### Step 3: Write the formula for the curved surface area
The formula for the curved surface area \( A \) of a cone is given by:
\[ A = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 4: Calculate the curved surface areas of both cones
For cone 1:
\[ A_1 = \pi r l_1 = \pi r (5k) = 5\pi rk \]
For cone 2:
\[ A_2 = \pi r l_2 = \pi r (4k) = 4\pi rk \]

### Step 5: Find the ratio of the curved surface areas
Now, we can find the ratio of the curved surface areas \( A_1 \) and \( A_2 \):
\[ \frac{A_1}{A_2} = \frac{5\pi rk}{4\pi rk} \]
The \( \pi \) and \( rk \) terms cancel out:
\[ \frac{A_1}{A_2} = \frac{5}{4} \]

### Conclusion
Thus, the ratio of the curved surface areas of the two cones is:
\[ \frac{A_1}{A_2} = \frac{5}{4} \]