Amelia runs a catering business. Based on her records, her weekly profit can be modeled by P=-2x^2+152x-150P=−2x 2 +152x−150, where xx is the number of meals she caters and PP is her profit.

The least number of meals Amelia needs to cater in order to begin making profit = 38 meals.

Step-by-step explanation:

Given - Amelia runs a catering business. Based on her records, her weekly  

            profit can be modeled by P= -2x² + 152x - 150, where x is the  

            number of meals she caters and P is her profit. When P is negative,  

            Amelia losses money.

To find - What is the least number of meals Amelia needs to cater in order  

              to begin making profit.

Proof -

As given, P = -2x² + 152x - 150

Now,

Above the break-even point, she only get profit

The break -even point is a point where there is no profit, and no loss.

Now,

For break-even point,

⇒

⇒-4x + 152 = 0

⇒- 4x = - 152

⇒4x = 152

⇒x = = 38

∴ we get

If Amelia caters at least 38 meals , then she makes Profit.