A video store rents, on average, 240 videos a day for $2.00 each. The store determined that for every $0.25 that it increases the rental fee, the number of daily rentals will decrease by 10. This relationship can be represented by
y=(240-10n)(2+0.25n) where y is the daily income in dollars from video rentals and n is the number of $0.25 increases. Based on this relationship, at what rental fee per video will the store have its highest daily income?
The daily income is represented by y, right? If you graph that function (substitute x for n to put it in your calculator), you’ll get a parabola. All you need to do is find the vertex! (Note that, in this problem, you’ll want to do some zooming out.)
Don’t want to graph? This is also a great one to backsolve on. Just put each answer choice in
for n and see which one gives you the biggest value for y!
Don’t want to do either of the easy ways? 🙂 Then you’re going to recognize that the equation represents a parabola, and then manipulate it so you can see its roots, and then take the average of the roots to find the n-value for the vertex. Note that doing this requires much more insight into the question than the previous solutions did.
So you’ve got roots at –8 and 24. The n-coordinate of the vertex will be smack dab in the middle of those. , so the vertex of the parabola is at n = 8.