Question

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. Assuming that any increase occurs in whole dollar amounts, what is the maximum possible increase that maintains the desired minimum revenue? Explain why this is true.

Answer 1

$3 max

Explanation:

Charge= b, Customer= c, Revenue= r

r= bc, currently, r= 16*10= $160

We know that: b+1 ⇒ c-2 and the target is r ≥ 130

So, this will all be reflected as:

b=10+x ⇒ c= 16-2x

• (10+x)(16-2x) ≥ 130
• 160 -20x +16x - 2x² ≥ 130
• -2x² - 4x + 30 ≥ 0
• x² + 2x -15 ≤ 0
• (x+1)² ≤ 4²
• x+1 ≤ 4 (negative value not considered)
• x ≤ 3

As we see the max increase amount is $3, when the revenue will be:

(10+3)*(16-3*2)= 13*10= $130