solving a word problem solving a system of linear equations of the form y=mx+b

Question

An air traffic controller is tracking two planes. To start, a plane A is at an altitute of 3000 feet and plane B is at an altitude of 1623 feet. Plane A is a gaining altitude at 40.25 feet per second and plane B is gaining altitude at 65.75 feet per second.

how many seconds will pass before the plane are at the same altitude?
what will their altitude be when they're at the same altitude?

Answer 1

The slope-intercept form of the equation of a line looks like:     y=mx + b

     where m is the slope and (0,b) is the y-intercept

      Note that the y-intercept is the value of y when x=0.

 

 

The problem gives important information:

  "To start"   allows to arbitrarily set t=0     (time = 0)

  "plane A is at an altitute of 3000 feet "      means   b₁ = 3000 ft

  "plane B is at an altitude of 1623 feet"      means   b₂ = 1623 ft

 

   [a little humor:  I was a pilot and measuring 1623 feet altitude is really precise!  this must be a math problem]

 

  "Plane A is a gaining altitude at 40.25 feet per second"      means    m₁ = 40.25 ft/sec

  "plane B is gaining altitude at 65.75 feet per second"      means   m₂ = 65.75 ft/sec

 

The equations for Altitude in terms of time are:

    A₁ = 40.25t₁ + 3000

    A₂ = 65.75t₂ + 1623

 

Now, to answer the questions:

  "how many seconds will pass before the plane are at the same altitude? "

            same altitude means A₁ = A₂   at t_s = t₁ = t₂         (t_s is "same time")

 

  So, set the equations equal and solve for t_s:

       A₁ = 40.25t_s + 3000 = A₂ = 65.75t_s + 1623

        40.25t_s + 3000 = 65.75t_s + 1623                    [re-write]

                  1377 = 20.50t_s

                     54 = t_s              [more humor:  amazing how this turns out to be a whole second!]

 

  "what will their altitude be when they're at the same altitude?"    means   PLZ reveal A_s  (that is, at t_s)

    You could use either equation (and you could check that they produce the same answer)

         A_s = 40.25t_s + 3000

         A_s = 40.25(54) + 3000

         A_s = 5173.5 ft       [more humor:  I was a pilot and you don't measure this accurately!]