Explain what the constant rate of change means in this context

Question

Stephen earns extra money babysitting he charges $31.25 for 5 hours and $50 for 8 hours

Answer 1

This is actually a fairly simple situation (i.e. Stephen charges $6.25 per hour).  However, the way it's presented is looking for you to think a little deeper.  Specifically, if a rate of change of one variable (cost) relative to another variable (time) is constant, then the function graphs a straight line and can be described by a linear equation.  Our goal then is to find the equation of cost (C) as a function of time (t).  Point slope format is a useful method of finding a linear equation.  First, we need to find slope.  Comparing the two given points, we have

 

ΔC / tΔ = (50 - 31.25) / (8 - 5) = 18.75 / 3 = 6.25

 

We could even state the slope as $6.25/hr if we wanted at this point.  Putting this into the point slope formula (using the first point), we have

 

(C - 31.25) / (t - 5) = 6.25

C - 31.25 = 6.25t - 31.25

C = 6.25t

 

So now we have an equation for C(t), that should make intuitive sense.

 

If you have any follow-on questions, let me know.