Chapter 1.4, Problem 18E

Reminder Round all answers to two decimal places unless otherwise indicated.

More on Profit This is a continuation of Exercises 15, 16, and 17. In this exercise, we use the formula for the total cost of the widget manufacturer found in Exercise $15$ and the formula for the total revenue found in Exercise $17$.

a. Use a formula to express the profit $P$ of this manufacturer as a function of $N$.

b. Consider the three production levels: $N=200$, $N=700$, and $N=1200$ . For each of these, determine whether the manufacturer has a loss, turns a profit, or is a break even point.

$15$.Total Cost The total cost $C$ for a manufacturer during a given time period is a function of the number $N$ of items produced during that period. To determine a formula for the total cost, we need to know the manufacturer’s fixed costs (covering things such as plant maintenance and insurance), as well as the cost for each unit produced, which is called the variable, cost. To find the total cost, we multiply the variable cost by the number of items produced during that period and then add the fixed costs.

Suppose that a manufacturer of widgets has fixed costs of $\$9000$ per month and that the variable cost is $\$15$ per widget (so it costs $\$15$ to produce $1$ widget).

a. Use a formula to express the total cost of this manufacturer in a month as a function of the number of widgets produced in a month. Be sure to state the units you use.

b. Express using functional notation the total cost if there are $250$ widgets produced in a month, and then calculate that value.

$16$.Total Revenue and Profit This is a continuation of Exercise 15. The total revenue $R$ for a manufacturer during a given time period is a function of the number $N$ of items produced during that period. To determine a formula for the total revenue, we need to know the selling price per unit of the item. To find the total revenue, we multiply this selling price by the number of items produced.

The profit $P$ for a manufacturer is the total revenue minus the total cost. If this number is positive, then the manufacturer turns a profit, whereas if this number is negative, then the manufacturer has a loss. If the profit is zero, then the manufacturer is at break-even point.

Suppose the manufacturer of widgets in Exercise 15 sells the widgets for $\$25\text{each}$.

a. Use a formula to express this manufacturer’s total revenue $R$ in a month as a function of the number of widgets produced in a month. Be sure to state the units you use.

b. Use a formula to express the profit $P$ of this manufacturer as a function of the number of widgets produced in a month. Be sure to state the units you use.

c. Express using functional notation the profit of this manufacturer if there are $250$ widgets produced in a month, and then calculate that value.

d. At the production level of $250$ widgets per month, does the manufacturer turn a profit or have a loss? What about at the production level of $1000$ widgets per month?

$17$.More on Revenue This is a continuation of Exercise $15$ and $16$. In general, the highest price $p$ per unit of an item which a manufacturer can sell $N$ items is not constant, but is rather a function of $N$. The total revenue $R$ is still the product of $p$ and $N$, but the formula for $R$ is more complicated when $p$ depends on $N$. Suppose the manufacturer of widgets in Exercises $15$ and Exercises $16$ no longer sells widgets for $\$25$ each. Rather, the manufacturer has developed the following table showing the highest price $p$, in dollars, of a widget at which $N$ widgets can be sold.

a. Verify that the formula $p=50-0.01N$ where $p$ is the price in dollars, give the same values as those in the table.

$N=Numberofwidgetssold$	$p=Price$
$100$	$49$
$200$	$48$
$300$	$47$
$400$	$46$
$500$	$45$

b. Use the formula from part a and the fact that $R$ is the product of $p$ and $N$ to find a formula expressing the total revenue $R$ as a function of $N$ for this widget manufacturer.

c. Express using functional notation the total revenue of this manufacturer if there are $450$ weights produced in a month, and then calculate that value.