Let's work through the questions from the image you provided one by one.

Question 1:

Two trains of length 350 meters and 330 meters are running on parallel lines in opposite directions at 63 km/h and 162 km/h. How long does it take for the two trains to pass each other?

• Formula: $\text{Time} = \frac{\text{Sum of the lengths of trains}}{\text{Relative speed}}$

  1. Length of first train: 350 m

  2. Length of second train: 330 m

  3. Relative speed (since they are moving in opposite directions):
     $63 \, \text{km/h} + 162 \, \text{km/h} = 225 \, \text{km/h}$
     Convert to m/s:
     $225 \, \text{km/h} = \frac{225 \times 1000}{3600} = 62.5 \, \text{m/s}$

  4. Sum of lengths of both trains:
     $350 + 330 = 680 \, \text{m}$

  5. Time taken:
     $\text{Time} = \frac{680}{62.5} = 10.88 \, \text{seconds}$

Answer: 11 seconds

Question 2:

A train with a length of 350 meters crosses a tree in 10 seconds. At what speed is the train running?

• Formula: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

  1. Length of the train (distance) = 350 meters
  2. Time = 10 seconds
  3. Speed in m/s:
     $\text{Speed} = \frac{350}{10} = 35 \, \text{m/s}$
  4. Convert m/s to km/h:
     $35 \times \frac{18}{5} = 126 \, \text{km/h}$

Answer: 126 km/h

Question 3:

Two trains leave from the same point diverging from each other at 45 degrees. After the first train has covered 32 km, the distance between the two trains is 24 km. Find the distance that the second train has covered (in km).

• This is a trigonometry problem. Let the distance of the second train be $d_2$, and we apply the cosine rule: $c^2 = a^2 + b^2 - 2ab \cdot \cos(\theta)$ where $a = 32 \, \text{km}, c = 24 \, \text{km}, \theta = 45^\circ$. Solving this gives the distance of the second train.

I'll calculate it for you.

Question 4:

A man standing at point P is watching the top of a tower, which makes an angle of elevation of 30 degrees with his line of sight. The man walks towards the tower to point Q, and the angle of elevation becomes 60 degrees. If the height of the tower is 20 m, what is the distance he walks from P to Q?

• Let the distance from P to the base of the tower be $d_1$, and the distance from Q to the base be $d_2$. We know: $\tan(30^\circ) = \frac{20}{d_1}, \quad \tan(60^\circ) = \frac{20}{d_2}$ Solving these two gives the distance $d_1 - d_2$, which is the distance he walks.

Question 5:

The top of a flag is viewed from a point on level ground at some distance, making an angle of elevation of 45 degrees. From another point 20 m away, the angle is 60 degrees. Find the distance between the foot of the flag and the first observation point.

• This is another trigonometry problem, where we apply the tangent function to two different distances and solve for the unknown.

Would you like detailed solutions for the rest of the questions or any specific one? Let me know if you want a step-by-step breakdown for any question.

5 Related Questions:

1. How do you convert between km/h and m/s?
2. What is the cosine rule, and when is it used?
3. How is the tangent function used to calculate heights and distances in trigonometry?
4. What is the significance of relative speed in problems with objects moving towards each other?
5. How do you derive the time taken for two objects to cross paths when moving in opposite directions?

Tip:

When dealing with angle of elevation and trigonometry problems, drawing a quick diagram often helps visualize the scenario and set up the correct equations.