Question

the function ( p(x) ) models the daily profit a bagel shop makes during its first week of business.

(a) how many days was the shop open before it started making more money than it was spending?

(b) how many days was the shop open before it started making a profit greater than $150 per day?

(c) determine the average rate of change in profit between day 2 and day 7. explain the meaning of your answer in terms of the problem situation.

Explanation:

Step1: Analyze when profit is positive (for part a)

The shop starts making more money than it spends when the profit \(p(x)>0\). Looking at the graph, the profit is non - negative ( \(p(x)\geq0\)) starting at \(x = 2\) (since for \(x<2\), \(p(x)\leq0\)).

Step2: Analyze when profit is greater than 150 (for part b)

We look for the value of \(x\) when \(p(x)>150\). From the graph, when \(y = 150\), we find the corresponding \(x\) value. The profit \(p(x)>150\) when \(x>5\).

Step3: Calculate the average rate of change (for part c)

The formula for the average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(a = 2\), \(b=7\). Let's assume \(p(2)=0\) (from the graph, at \(x = 2\), \(y = 0\)) and \(p(7)=200\) (from the graph, at \(x = 7\), \(y = 200\)). Then the average rate of change is \(\frac{p(7)-p(2)}{7 - 2}=\frac{200 - 0}{5}\).

Answer:

a. \(2\) days.
b. \(5\) days.
c. The average rate of change is \(\frac{200-0}{5}=40\) dollars per day. This means that, on average, the daily profit of the bagel shop increased by \(40\) dollars each day from Day \(2\) to Day \(7\).