Question

In this activity, you will create quadratic inequalities in one variable and use them to solve problems. Read this scenario, and then use the information to answer the questions that follow.
Noah manages a buffet at a local restaurant. He charges $10 for the buffet. On average, 16 customers choose the buffet as their meal every hour. After surveying several customers, Noah has determined that for every $1 increase in the cost of the buffet, the average number of customers who select the buffet will decrease by 2 per hour. The restaurant owner wants the buffet to maintain a minimum revenue of $130 per hour.

Noah wants to model this situation with an inequality and use the model to help him make the best pricing decisions.

Answer 1

Answer:

  A: cost = 10+x; customers = 16-2x

  B: -2x² -4x +160 ≥ 130

Step-by-step explanation:

In these price and revenue optimization problems, the relation between price and sales is given in the problem statement. After that is translated to math terms, the revenue as the product of price and sales can be determined.

Part A

The variable x is defined as the number of $1 increases from a price of $10. That means ...

  Cost = 10 +x

For each $1 increase, the number of customers decreases by 2 from a starting value of 16:

  Customers = 16 -2x

__

Part B

The problem statement tells you the revenue (per hour) is the product of cost and number of customers. The owner wants this to be at least 130:

  (Cost)(Customers) ≥ 130

  (10 +x)(16 -2x) ≥ 130

  160 -20x +16x -2x² ≥ 130 . . . . eliminate parentheses

  -2x² -4x +160 ≥ 130 . . . . . . . . . put in required standard form

_____

Additional comment

The attached graph shows the solution to this quadratic inequality is ...

  -5 ≤ x ≤ 3

and that revenue can be increased by $2 per hour by decreasing cost $1 (x=-1).