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Max/Min Problem – Maximizing Revenue, Selling Calculators

Q: Calculators are sold to students for 20 dollars each. Three hundred students are willing to buy them at that price. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. What selling price will produce the maximum revenue and what will the maximum revenue be?

A: You have to use the Revenue equation:

R = (Selling price) * (Number Sold)

When the selling price is $20, three hundred students are willing to buy them.

R = (20)*(300)

But for every 5 dollar increase in price (+5x), there is a 30 student decrease (-30x):

R = (20+5x)(300-30x)

TI-Nspire Graphing Calculator

This is the beautiful TI-Nspire Color Graphing calculator. Graphing calculators make problems like these easy to visualize.

You can expand this out and complete the square to turn it into Parabolic form, or you can use Wolfram Alpha to graph the parabola. In class, you can use a graphing calculator. This can help a lot on a test, as most teachers probably won’t let you use Wolfram Alpha in class. [Note: In this problem we’re just exploring the different prices people would pay for what is likely a scientific calculator (based on the $20 to $35 price range) and that graphing calculators are much more expensive! Check out our guide which contains a graphing calculator comparison for our top 5 choices.]

 

If you do it the long way (i.e. by hand – which you should be able to do as well!) the steps are:

R = (20+5x)(300-30x)
R= 600 -600x +1500x -150x^2
R = -150x^2 + 900x + 600

Complete the square here to find:

R = -150(x-3)^2 + 7350

So you find that the max revenue is $7350, and it happens when x=3, which would bring the price up to $35 instead of $20. And they would have 210 sales, instead of 300. While they might not actually work out the quadratic function to come up with a precise number, managers at movie theaters, bus companies, airlines, video game companies, and so on use reasoning like this all the time.

Note the x=3 and the Max Revenue of $7350, and see how similar it is to an area question, or a regular question with y and x.

Hope this helps. If you need help completing the square, let me know.