Chapter 3.1, Problem 78E

(a)

To determine

To find: The function that models weekly profit in terms of price per feeder.

Answer to Problem 78E

The function that models weekly profit in terms of price per feeder is equal to $P\left(x\right)=80-8x$ .

Explanation of Solution

Given information:

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $\$6$ , and the society sells an average of $20$ per week at a price of $\$10$ each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses $2$ sales per week.

Calculation:

As given a feeder costs $\$6$ and sold at $\$10$ . So, the profit for each feeder is $\$10-\$6=\$4$ .

Assume that the society increases the price by $x$ dollars, then the society loses $2x$ sales per week.

The total sale of the society per week is equal to $20-2x$ .

So, the total profit per week for the society is equal to $4\left(20-2x\right)=80-8x$ .

Therefore, the function that models weekly profit in terms of price per feeder is equal to $P\left(x\right)=80-8x$ .

(b)

To determine

To find: The price that society should charge for each feeder to maximize the profit and also the maximum weekly profit.

Answer to Problem 78E

The price that society should charge for each feeder to maximize the profit is equal to $\$20$ and the maximum weekly profit is equal to $\$80$ .

Explanation of Solution

Given information:

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $\$6$ , and the society sells an average of $20$ per week at a price of $\$10$ each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses $2$ sales per week.

Calculation:

As calculated in part(a), the profit function is given by $P\left(x\right)=80-8x$ .

To maximize the profit, graph the profit function.

To graph a function $y=80-8x$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=80-8x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{80}\boxed{-}\boxed{8}\boxed{\text{X}}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=80-8\text{X}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure (1)

As observed from the graph, the profit function has maximum value equal to $80$ at $x=0$ .

The price for each feeder is equal to $\$10-x=\$10$ .

Therefore, the price that society should charge for each feeder to maximize the profit is equal to $\$20$ and the maximum weekly profit is equal to $\$80$ .