A circular curve of a highway is designed for traffic moving at 72 km/h. if the radius of the curved path is 100 m, the correct angle of banking of the road should be given by:

Options

A. `tan^(-1)((2)/(5))`
B. `tan^(-1)((3)/(5))`
C. `tan^(-1)((1)/(5))`
D. `tan^(-1)((1)/(4))`

Explanation

To solve the problem of finding the correct angle of banking of the road for a circular curve designed for traffic moving at 72 km/h with a radius of 100 m, we can follow these steps:

### Step 1: Convert the speed from km/h to m/s
The speed given is 72 km/h. To convert this speed into meters per second (m/s), we use the conversion factor:

\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]

Calculating:

\[ \text{Speed in m/s} = 72 \times \frac{5}{18} = 20 \text{ m/s} \]

### Step 2: Use the formula for banking angle
The formula for the tangent of the banking angle (\(\theta\)) for safe turning is given by:

\[ \tan(\theta) = \frac{V^2}{Rg} \]

Where:
- \(V\) is the speed (20 m/s),
- \(R\) is the radius of the curve (100 m),
- \(g\) is the acceleration due to gravity (approximately \(10 \text{ m/s}^2\)).

### Step 3: Substitute the values into the formula
Now we substitute the values into the formula:

\[ \tan(\theta) = \frac{(20)^2}{100 \times 10} \]

Calculating:

\[ \tan(\theta) = \frac{400}{1000} = 0.4 \]

### Step 4: Calculate the angle \(\theta\)
To find the angle \(\theta\), we take the arctangent (inverse tangent) of 0.4:

\[ \theta = \tan^{-1}(0.4) \]

Using a calculator or trigonometric tables, we find:

\[ \theta \approx 21.8^\circ \]

### Step 5: Conclusion
Thus, the correct angle of banking of the road should be approximately \(21.8^\circ\).

---