How do I answer these?

Question

An office building owner agrees to buy a minimum of 250 chairs and up to 430 chairs from a supplier. The price will be $88 per chair if only 250 chairs are bought, but will be discounted by $0.25 per chair (on the entire order) for every chair ordered in addition to the minimum. Answer the questions below, rounding your answers to the nearest whole dollar. a) What is the largest revenue the supplier can make under this deal? revenue = $ b) What is the least revenue the supplier can make under this deal? revenue = $

Answer 1

Hi Makenzie K.,

You solve this problem just as you would solve any other word problem. You turn it into a math equation. You do appreciate that math equations can be solved, but word problems just run around in your head forever, right?

So, your goal is to do that. The target is "revenue", which appears here to refer to the sales price on the order. Revenue could also have other meanings, depending on the circumstances. But since the problem doesn't clue you in, assume the easiest case. Always!

So how much is the deal?

So you need something you can latch onto to make an equation out of. Since revenue is the desired target, go the "other way" in the problem, to "number of chairs". (So there's a lesson here. If my desired answer is one thing, always start with something else to build an equation with! You can't start with what you want to end up with, and expect to get any place!)

So, since 250 chairs is the minimum order. Let's start "there", and add chairs to that, call the number of additional chairs "x".

Now the tricky part comes. We must set up the revenue, based on "x" as the input, using all the info supplied. OK, what is that?

The 250 chairs were $88 per chair, so that is $22,000 right there. Next, for each additional chair (and there are "x" of them"), the price per chair comes down by $0.25, so that it is only (88-0.25x) per chair. Now let's see how that plays out into total cost! First of all the initial 250 chairs are no longer being charged $88 each (the problem said "on the entire order"). So the easiest way to deal with that is as a "rebate" on them, at -250(0.25x). So, for example, for x=10 additional chairs, the rebate on those first 250 chairs would be $2.50/chair; extended over 250 chairs, that would be $625 (rebate). Fine so far. Also, the cost ON those additional "x" chairs would be x*(88-0.25x).

So, the total charge, given a total order of 250 chairs + x, would be: y = 22000-250(0.25x) + x(88-0.25x) . Do you see how we went through methodically to get the little pieces of charge in the entire order?

Time to expand! y = 22000-62.5x + 88x - 0.25x^2 = -0.25x^2 + 25.5x + 22000 .

At that point, you immediately take a step back, and say, time to bring in calculus! The first derivative of this revenue function, if it has a zero, will have a local minimum or maximum there. So, take the first derivative:

y' = -0.50x + 25.5 ; set to zero, and solve: x = 51 is either a local minimum or maximum. I'll leave it to you to use the revenue function to figure the revenue at x=0 or (430-250), and pick the appropriate value for your other of maximum or minimum revenue.

--Cheers, --Mr. d.