The rocket's can only burn for 15 minutes and then must be released. Each Rocket also weighs 10,000 lbs adding to the total weight of the craft.
For each engine you need an additional 10,000 lbs of thrust. (We are assuming that the weight of the engine itself is minimal.)
My issue with their answer is the following:
Taken the scenario they have played out the thrust to weight ratio's would be as follows.
stage 1: 180,000 lbs weight to 320,000 lbs thrust or 1:1.7
stage 2: 100,000 lbs weight to 160,000 lbs thrust or 1:1.6
stage 3: 60,000 lbs weight to 80,000 lbs thrust or 1:1.33
stage 4: 40,000 lbs weight to 40,000 lbs thrust or 1:1
If the rocket stays in the earths atmosphere it will stop traveling at the final stage because it has a 1:1 thrust to weight ratio. (just enough thrust to keep it from falling backwards to the ground)
If the rocket passes out of the earth atmosphere and gravitational pull the vacuum of space will make propulsion far less taxing and the vehicle will be able to fly with much less power.