Help with max rocket height. Heights differ with methods.

[Breadsticks]

Homework Statement

A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 30 m/s^2 for 30s, then runs out of fuel. Ignore any air resistance effects. What is the rocket’s maximum altitude?

Homework Equations

Δx=V₀t+(1/2)at²
V=V₀+at
2aΔx=V²-V₀²

The Attempt at a Solution

Δx=(1/2)(30m/s²)(30s)²=13500m

V=0(m/s)+(1/2)(30m/s²)(30s)=900m/s
2(-9.81m/s²)Δx=0²-900²m/s
=> Δx=41300m

Total height: 55000m

However, I did it another way and received a different result.
V=V₀+at
0=900+(-9.81)t
t=91.7s

Δx=(900m/s)(91.7s)=82530m
Total height: 96000m

Why is this?

 

[Titan97]

In your second method, there is a missing ##-\frac{1}{2}gt^2## term. After it runs out of fuel, its acceleration will be g but not 0.