If the demand for a pair of shoes is given by $2 p+5 q=200$ and the supply function for it is $p-2 q=10$, compare the quantity demanded and the quantity supplied when the price is $\$ 60$. Will there be a surplus or shortfall at this price?

A personal computer retail company sells five different computer models through three stores. The inventory of each model on hand in each store is summarized in matrix M. Wholesale (W) and retail (R) values of each model computer are summarized in matrix N.

$$\begin{matrix} \text{A B C D E}\\ \text{}{M=\left[\begin{array}{rrrrr}{4} & {2} & {3} & {7} & {1} \\ {2} & {3} & {5} & {0} & {6} \\ {10} & {4} & {3} & {4} & {3}\end{array}\right]} & \text{}{\begin{array}{l}{\text { Store } 1} \\ {\text { Store } 2} \\ {\text { Store } 3}\end{array}}\\ \end{matrix}$$

$$\begin{matrix} \text{W R} & \text{} & \text{}\\ \text{}{N=\left[\begin{array}{cc}{\$ 700} & {\$ 840} \\ {\$ 1,400} & {\$ 1,800} \\ {\$ 1,800} & {\$ 2,400} \\ {\$ 2,700} & {\$ 3,300} \\ {\$ 3,500} & {\$ 4,900}\end{array}\right]} & \text{}{\begin{array}{l}{A} \\ {B} \\ {C} \\ {D} \\ {E}\end{array}} & \text{Model}\\ \end{matrix}$$

																																																																																																																																																																																																																																																																																																																																																													 (A) What is the retail value of the inventory at store 2? (B) What is the wholesale value of the inventory at store 3? (C) If either product MN or NM has a meaningful interpretation, find the product and label its rows and columns. What do the entries represent? (D) Discuss methods of matrix multiplication that can be used to find the total inventory of each model on hand at all three stores. State the matrices that can be used and perform the necessary operations.

Electronic equipment manufacturer Dynamo Electric Inc. makes several types of surge protectors. Their base model surge protector has monthly fixed costs of $\$ 1045$. This particular model wholesales for $\$ 10$ each and costs $\$ 4.50$ per unit to manufacture.

(a) Write the function for Dynamo's monthly total costs.

(b) Write the function for Dynamo's monthly total revenue.

( c ) Write the function for Dynamo's monthly profit.

(d) Find the number of this type of surge protector that Dynamo must produce and sell each month to break even.

A manufacturer sells watches for $\$ 50$ per unit. The fixed cost related to this product is $\$ 10,000$ per month, and the variable cost per unit is $\$ 30$.

(a) Write the equations of the revenue and cost functions.

(b) How many watches must be sold to break even?

A manufacturer sells belts for $\$ 12$ per unit. The fixed costs are $\$ 1600$ per month, and the variable cost per unit is $\$ 8$.

(a) Write the equations of the revenue and cost functions.

(b) Find the break-even point.

A jewelry maker incurs costs for a necklace according to

$$C(x)=35 x+1650$$

If the revenue function for the necklaces is

$$R(x)=85 x$$

how many necklaces must be sold to break even?

Suppose a ceiling fan manufacturer has the total cost function $C(x)=43 x+1850$ and the total revenue function $R(x)=80 x$.

(a) What is the equation of the profit function for this commodity?

(b) What is the profit on $30$ units? Interpret your result.

( c ) How many fans must be sold to avoid losing money?

A manufacturer of shower surrounds has a revenue function of

$$R(x)=81.50 x$$

and a cost function of

$$C(x)=63 x+1850$$

Find the number of units that must be sold to break even.

A manufacturer of DVD players has monthly fixed costs of $\$ 9800$ and variable costs of $\$ 65$ per unit for one particular model. The company sells this model to dealers for $\$ 100$ each. (a) For this model DVD player, write the function for monthly total costs. (b) Write the function for total revenue. (c) Write the function for profit. (d) Find $C(250), R(250)$, and $P(250)$ and interpret each answer. (e) Find $C(400), R(400)$, and $P(400)$ and interpret each answer. (f) Find the marginal profit and write a sentence that explains its meaning.

Suppose a company has fixed costs of $\$ 28,000$ and variable cost per unit of $\frac{2}{5} x+222$ dollars, where $x$ is the total number of units produced. Suppose further that the selling price of its product is $1250-\frac{3}{5} x$ dollars per unit.

(a) Find the break-even points.

(b) Find the maximum revenue.

( c ) Form the profit function from the cost and revenue functions and find maximum profit.

(d) What price will maximize the profit?

Extreme Protection Inc. manufactures helmets for skiing and snow boarding. The fixed costs for one model of helmet are $\$ 6600$ per month. Materials and labor for each helmet of this model are $\$ 35$, and the company sells this helmet to dealers for $\$ 60$ each.

(a) For this helmet, write the function for monthly total costs.

(b) Write the function for total revenue.

( c ) Write the function for profit.

(d) Find $C(200), R(200)$, and $P(200)$ and interpret each answer.

(e) Find $C(300), R(300)$, and $P(300)$ and interpret each answer.

(f) Find the marginal profit and write a sentence that explains its meaning.

A company charting its profits notices that the relationship between the number of units sold, $x$, and the profit, $P$, is linear. If $200$ units sold results in $\$ 3100$ profit and $250$ units sold results in $\$ 6000$ profit, write the profit function for this company. Find the marginal profit.

Given $C(x)=21.95 x+1400$ and $R(x)=20 x$, find the profit function.

(a) What is the marginal profit, and what does it mean?

(b) What should a firm with these cost, revenue, and profit functions do? (Hint: Graph the profit function and see where it goes.)

A linear revenue function is $R=38.95 x$.

(a) What is the slope?

(b) What is the marginal revenue, and what does it mean?

( c ) What is the revenue received from selling one more item if $50$ are currently being sold? If $100$ are being sold?