To find the maximum height of the projectile, we need to analyze its motion and use the kinematic equations. Let's break down the problem step by step:

1. Initial Velocity Components:
The given velocity u makes an angle θ with the vertical. We can resolve the initial velocity u into its vertical and horizontal components as follows:
- Vertical component: u * sin(θ)
- Horizontal component: u * cos(θ)

2. Time of Flight:
The projectile crosses the top of the first pole after 1 second and the top of the second pole after 3 seconds. The time of flight (T) is the total time taken by the projectile to reach maximum height and then come back down. Therefore, T = 3 - 1 = 2 seconds.

3. Maximum Height:
To find the maximum height (H) of the projectile, we can use the equation for vertical displacement:
H = (u * sin(θ)) * T - (1/2) * g * T^2
- (u * sin(θ)) is the initial vertical velocity component
- T is the time of flight
- (1/2) * g * T^2 is the gravitational vertical displacement

Substituting the values, we have:
H = (u * sin(θ)) * 2 - (1/2) * (9.8 m/s^2) * (2 s)^2
H = 2u * sin(θ) - 19.6 m

4. Simplification:
We know that the projectile crosses the top of each pole, which means its height at that point is equal to the height of the pole (h). Therefore, at t = 1 s, the height is h and at t = 3 s, the height is also h.

Using the equation for vertical displacement, we can equate the height of the projectile at t = 1 s and t = 3 s to the height of the poles:
h = (u * sin(θ)) * 1 - (1/2) * (9.8 m/s^2) * (1 s)^2
h = u * sin(θ) - 4.9 m

h = (u * sin(θ)) * 3 - (1/2) * (9.8 m/s^2) * (3 s)^2
h = 3u * sin(θ) - 44.1 m

5. Solving the Equations:
We have two equations:
h = u * sin(θ) - 4.9 m
h = 3u * sin(θ) - 44.1 m

By subtracting the first equation from the second equation, we eliminate h:
3u * sin(θ) - u * sin(θ) = 44.1 m - 4.9 m
2u * sin(θ) = 39.2 m

Finally, dividing both sides by 2sin(θ), we get:
u = 39.2 m / (2sin(θ))

Substituting this value of u back into the equation for maximum height, we have:
H = 2u * sin(θ) - 19.6 m
H = 2 * (39.2 m / (2sin(θ))) * sin(θ) - 19.6