Chapter 4.8, Problem 35E

Solution

a.

The month in which the shop first began to make money.

The month in which the shop first began to make money is March

Given:

Given, owing to startup costs and seasonal variations, Gina found that the monthly profit in her bagel shop during the first year followed an up-and-down pattern that could be modeled by $P=2t-7\sin \left(\pi t/3\right)$ , where P was measured in hundreds of dollars and t was measured in months after January 1.

Calculation:

For the shop to make profit, $P>0$ i.e., when the graph of the function is above the x -axis.

Graphing the function:

  

From the graph, it is seen that the shop starts making Profit between 2 and 3 i.e., in March.

Conclusion:

The month in which the shop first began to make money is March

b.

The month in which the shop enjoyed its greatest profit in that first year.

The month in which the shop enjoyed its greatest profit in that first year is November.

Given:

Given, owing to startup costs and seasonal variations, Gina found that the monthly profit in her bagel shop during the first year followed an up-and-down pattern that could be modeled by $P=2t-7\sin \left(\pi t/3\right)$ , where P was measured in hundreds of dollars and t was measured in months after January 1.

Calculation:

Graphing the function:

  

From the graph, in the first year i.e., between 0 and 12, the highest profit is made between 10 and 11. So the highest profit is made in November.

Conclusion:

The month in which the shop enjoyed its greatest profit in that first year is November.