It was calculated that a shell when fired from a gun with a certain velocity and at an angle of elevation `5pi//36` radius should strike a given target. In actual practice it was found that a hill just intervened in the trajectory. At what angle of elevation should the gun be fired to hit the target ?

Options

A. `(5 pi)/(36) radius`
B. `(7 pi)/(36) radius`
C. `(11 pi)/(36) radius`
D. `(13 pi)/(36) radius`

Explanation

To solve the problem, we need to find the new angle of elevation at which the gun should be fired to hit the target, given that a hill intervenes in the trajectory.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- The initial angle of elevation is given as \( \theta_1 = \frac{5\pi}{36} \) radians. This angle was calculated to hit the target without any obstacles.

2. **Identifying the Effect of the Hill**:
- The presence of the hill means that the projectile must be fired at a higher angle to ensure it clears the hill and still reaches the target.

3. **Using the Concept of Complementary Angles**:
- For a given range, there are generally two angles of projection that can achieve the same horizontal distance (range). If one angle is \( \theta \), the other angle is \( 90^\circ - \theta \) (or \( \frac{\pi}{2} - \theta \) in radians).

4. **Calculating the New Angle**:
- Since the original angle of projection is \( \theta_1 = \frac{5\pi}{36} \), the complementary angle (which will be the new angle of projection to hit the target while clearing the hill) is calculated as:
\[ \theta_2 = \frac{\pi}{2} - \theta_1 \]
- Substituting the value of \( \theta_1 \):
\[ \theta_2 = \frac{\pi}{2} - \frac{5\pi}{36} \]

5. **Finding a Common Denominator**:
- To perform the subtraction, convert \( \frac{\pi}{2} \) into a fraction with a denominator of 36:
\[ \frac{\pi}{2} = \frac{18\pi}{36} \]
- Now substitute this back into the equation for \( \theta_2 \):
\[ \theta_2 = \frac{18\pi}{36} - \frac{5\pi}{36} = \frac{(18 - 5)\pi}{36} = \frac{13\pi}{36} \]

6. **Conclusion**:
- The new angle of elevation at which the gun should be fired to hit the target while clearing the hill is:
\[ \theta_2 = \frac{13\pi}{36} \text{ radians} \]

### Final Answer:
The angle of elevation should be \( \frac{13\pi}{36} \) radians.

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