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?

SAT2 question:)

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.

    \begin{align*} y&=(240-10n)(2+0.25n)\\ y&=10(24-n)\frac{1}{4}(8+n)\\ y&=2.5(24-n)(8+n) \end{align*}

So you’ve got roots at –8 and 24. The n-coordinate of the vertex will be smack dab in the middle of those. \dfrac{-8+24}{2}=8, so the vertex of the parabola is at n = 8.

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