You found the area for time 4-5s wrong.
You found the red part, but not the blue part. The blue part is a rectangle.
Also, I think for this problem you can assume displacement and distance are meant to be the same thing.
How do you know that the angle of the friction is 180? Wouldn't the angle of friction be 240 degrees, because of the 60 degree incline?
How did you draw your FBD?
You first have to find the height it reaches in the first 20 seconds. Vf will not be zero after 20 seconds. After 20 seconds it is coasting (like a car when you take your foot off of the gas). It will take more time for it to reach 0 m/s (maximum height).
Try solving the coasting part on your...
Why did you use g for acceleration during the first 20 seconds? The rocket is accelerating at +10m/s2 during the first 10 seconds.
You would use g for acceleration after the 20 seconds, since that's when the motors are off.
Nevermind, gneill's suggestion is better.
You know initial velocity, time, and acceleration (gravity, as gneill pointed out). You can solve for final velocity, which will allow you to then solve for distance.
Does that make sense?