Keeping the banking angle same, to increase the maximum speed with which a vehicle can travel on the curved road by 10%, the radius of curvature of the road has to be changed from 20 m to

Options

A. 16 m
B. 18 m
C. 24.2 m
D. 30.5 m

Explanation

To solve the problem of increasing the maximum speed of a vehicle traveling on a curved road while keeping the banking angle the same, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The maximum speed \( V \) of a vehicle on a banked curve is given by the formula:
\[ V = \sqrt{g \cdot R \cdot \tan(\theta)} \]
where \( g \) is the acceleration due to gravity, \( R \) is the radius of curvature, and \( \theta \) is the banking angle.

2. **Initial Conditions**: We are given that the initial radius of curvature \( R_1 = 20 \, m \) and we want to increase the speed by 10%. Thus, if the initial speed is \( V_1 \), the new speed \( V_2 \) will be:
\[ V_2 = 1.1 \cdot V_1 \]

3. **Setting Up the Equation**: Since the banking angle remains the same, we can set up the relationship between the initial and new conditions:
\[ \frac{V_2^2}{R_2} = \frac{V_1^2}{R_1} \]
This implies:
\[ R_2 = R_1 \cdot \left(\frac{V_2}{V_1}\right)^2 \]

4. **Substituting Values**: Now, substituting \( R_1 = 20 \, m \) and \( V_2 = 1.1 \cdot V_1 \):
\[ R_2 = 20 \cdot \left(\frac{1.1 \cdot V_1}{V_1}\right)^2 \]
Simplifying this gives:
\[ R_2 = 20 \cdot (1.1)^2 \]

5. **Calculating \( R_2 \)**: Now calculate \( (1.1)^2 \):
\[ (1.1)^2 = 1.21 \]
Therefore:
\[ R_2 = 20 \cdot 1.21 = 24.2 \, m \]

### Final Answer:
The new radius of curvature \( R_2 \) required to increase the maximum speed by 10% while keeping the banking angle the same is:
\[ \boxed{24.2 \, m} \]