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### Like us? # 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) 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.