How many seconds will pass before the planes are at the same altitude? What will their altitude be when they're at the same altitude?

Question

An air traffic controller is tracking two planes. To start, Plane A is at an altitude of 3500 feet and Plane B is at an altitude of 1512 feet. Plane A is gaining altitude at 30.25 feet per second and Plane B is gaining altitude at 65.75 feet per second.How many seconds will pass before the planes are at the same altitude? What will their altitude be when they're at the same altitude?

Answer 1

Create a linear equation for the motion of each plane. Let "y" by=e the altitude and let "x" be the time in seconds:

Plane A: y = 30.25x + 3500 (when x = 0, the altitude is 3500 and the plane is gaining 30.25 feet every second)

Plane B: y = 65.75x + 1512 (when x = 0, the altitude is 1512 and the plane is gaining 65.75 feet every second)

You want to know about the point where they both have the same altitude, meaning the two "y" values are equal

Well if y = 30.25x + 3500 and y = 65.75x + 1512 and the two y's are the same then:

30.25x + 3500 = 65.75x + 1512

3500 - 1512 = 65.75x - 30.25x

1988 = 35.5x

x = 1988/35.5 = 56 meaning 56 seconds pass before the planes are at the same altitude.

If x = 56, then plug that value of "x" into either equation:

y = 30.25x + 3500 so:

y = 30.25(56) + 3500 = 5194 so the altitude when both planes have the same altitude is 5194 feet