A company has cost function C(q) = 4000+2q dollars and revenue function R(q) = 10q

In Figure 1.54, which shows the cost and revenue functions for a product, label each of the following: (a) Fixedcosts (b) Break-evenquantity (c) Quantities at which the company: (i) Makes a profit (ii) Loses money q $ C R Figure 1.54

Figure 1.55 shows cost and revenue for a company. (a) Approximately what quantity does this company have to produce to make a profit? (b) Estimate the profit generated by 600 units. 100 300 500 700 900 500 1500 2500 C(q) R(q) q (quantity) $ Figure 1.55

(a) Estimate the fixed costs and the marginal cost for the cost function in Figure 1.56. (b) Estimate C(10) and interpret it in terms of cost. 5 10 15 20 25 30 100 200 300 C(q) q $ Figure 1.56

Values of a linear cost function are in Table 1.27. What are the fixed costs and the marginal cost? Find a formula for the cost function. Table 1.27 q 0 5 10 15 20 C(q) 5000 5020 5040 5060 5080

The cost C, in millions of dollars, of producing q items is given by C = 5.7 + 0.002q. Interpret the 5.7 and the 0.002 in terms of production. Give units.

(a) Give an example of a possible company where the fixed costs are zero (or very small). (b) Give an example of a possible company where the marginal cost is zero (or very small).

Suppose that q = f(p) is the demand curve for a product, where p is the selling price in dollars and q is the quantity sold at that price. (a) What does the statement f(12) = 60 tell you about demand for this product? (b) Do you expect this function to be increasing or decreasing? Why?

A company has cost and revenue functions, in dollars, given by C(q) = 6000+10q and R(q) = 12q. (a) Find the cost and revenue if the company produces 500 units. Does the company make a profit? What about 5000 units? (b) Find the break-even point and illustrate it graphically.

The demand curve for a quantity q of a product is q = 5500 100p where p is price in dollars. Interpret the 5500 and the 100 in terms of demand. Give units. 1

A demand curve is given by 75p + 50q = 300, where p is the price of the product, in dollars, and q is the quantity demanded at that price. Find p- and q-intercepts and interpret them in terms of consumer demand. 1

An online seller of T-shirts pays $500 to start up the website and $6 per T-shirt, then sells the T-shirts for $12 each. 1

A car wash operator pays $35,000 for a franchise, then spends $10 per car wash, which costs the consumer $15. 1

A couple running a house-cleaning business invests $5000 in equipment, and they spend $15 in supplies to clean a house, for which they charge $60. 1

A lemonade stand operator sets up the stand for free in front of the neighbors house, makes 5 quarts of lemonade for $4, then sells each 8-oz cup for 25 cents. 1

A company thatmakes Adirondack chairs has fixed costs of $5000 and variable costs of $30 per chair. The company sells the chairs for $50 each. (a) Find formulas for the cost and revenue functions. (b) Find the marginal cost and marginal revenue. (c) Graph the cost and the revenue functions on the same axes. (d) Find the break-even point. 1

An amusement park charges an admission fee of $21 per person as well as an additional $4.50 for each ride. (a) For one visitor, find the parks total revenue R(n) as a function of the number of rides, n, taken. (b) Find R(2) and R(8) and interpret your answers in terms of amusement park fees. 1

A photocopying company has two different price lists. The first price list is $100 plus 3 cents per copy; the second price list is $200 plus 2 cents per copy. (a) For each price list, find the total cost as a function of the number of copies needed. (b) Determine which price list is cheaper for 5000 copies. (c) For what number of copies do both price lists charge the same amount? 1

A company has cost function C(q) = 4000+2q dollars and revenue function R(q) = 10q dollars. (a) What are the fixed costs for the company? (b) What is the marginal cost? (c) What price is the company charging for its product? (d) Graph C(q) and R(q) on the same axes and label the break-even point, q0. Explain how you know the company makes a profit if the quantity produced is greater than q0. (e) Find the break-even point q0. 1

A movie theater has fixed costs of $5000 per day and variable costs averaging $6 per customer. The theater charges $11 per ticket. (a) How many customers per day does the theater need in order to make a profit? (b) Find the cost and revenue functions and graph them on the same axes. Mark the break-even point. 2

A company producing jigsaw puzzles has fixed costs of $6000 and variable costs of $2 per puzzle. The company sells the puzzles for $5 each. (a) Find formulas for the cost function, the revenue function, and the profit function. (b) Sketch a graph of R(q) and C(q) on the same axes. What is the break-even point, q0, for the company? 2

Production costs for manufacturing running shoes consist of a fixed overhead of $650,000 plus variable costs of $20 per pair of shoes. Each pair of shoes sells for $70. (a) Find the total cost, C(q), the total revenue, R(q), and the total profit, (q), as a function of the number of pairs of shoes produced, q. (b) Find the marginal cost, marginal revenue, and marginal profit. (c) How many pairs of shoes must be produced and sold for the company to make a profit? 2

A $15,000 robot depreciates linearly to zero in 10 years. (a) Find a formula for its value as a function of time. (b) How much is the robot worth three years after it is purchased? 2

A $50,000 tractor has a resale value of $10,000 twenty years after it was purchased. Assume that the value of the tractor depreciates linearly from the time of purchase. (a) Find a formula for the value of the tractor as a function of the time since it was purchased. (b) Graph the value of the tractor against time. (c) Find the horizontal and vertical intercepts, give units, and interpret them. 2

A new bus worth $100,000 in 2010 depreciates linearly to $20,000 in 2030. (a) Find a formula for the value of the bus, V , as a function of time, t, in years since 2010. (b) What is the value of the bus in 2015? (c) Find and interpret the vertical and horizontal intercepts of the graph of the function. (d) What is the domain of the function? 2

A corporate office provides the demand curve in Figure 1.57 to its ice cream shop franchises. At a price of $1.00 per scoop, 240 scoops per day can be sold. (a) Estimate how many scoops could be sold per day at a price of 50c/ per scoop. Explain. (b) Estimate how many scoops per day could be sold at a price of $1.50 per scoop. Explain. 2

The table shows the cost of manufacturing various quantities of an item and the revenue obtained from their sale. (b) Calculate the profit or loss for each of the quantities shown. Estimate the most profitable production level. 2

One of Tables 1.28 and 1.29 represents a supply curve; the other represents a demand curve. (a) Which table represents which curve? Why? (b) At a price of $155, approximately how many items would consumers purchase? (c) At a price of $155, approximately how many items would manufacturers supply? (d) Will the market push prices higher or lower than $155? (e) What would the price have to be if you wanted consumers to buy at least 20 items? (f) What would the price have to be if you wanted manufacturers to supply at least 20 items? Table 1.28 p ($/unit) 182 167 153 143 133 125 118 q (quantity) 5 10 15 20 25 30 35 Table 1.29 p ($/unit) 6 35 66 110 166 235 316 q (quantity) 5 10 15 20 25 30 35 2

Figure 1.58 shows supply and demand for a product. (a) What is the equilibrium price for this product? At this price, what quantity is produced? (b) Choose a price above the equilibrium pricefor example, p = 12. At this price, how many items are suppliers willing to produce? How many items do consumers want to buy? Use your answers to these questions to explain why, if prices are above the equilibrium price, the market tends to push prices lower (toward the equilibrium). (c) Now choose a price below the equilibrium price for example, p = 8. At this price, how many items are suppliers willing to produce? How many items do consumers want to buy? Use your answers to these questions to explain why, if prices are below the equilibrium price, the market tends to push prices higher (toward the equilibrium). 2

A company produces and sells shirts. The fixed costs are $7000 and the variable costs are $5 per shirt. (a) Shirts are sold for $12 each. Find cost and revenue as functions of the quantity of shirts, q. (b) The company is considering changing the selling price of the shirts. Demand is q = 2000 40p, where p is price in dollars and q is the number of shirts. What quantity is sold at the current price of $12? What profit is realized at this price? (c) Use the demand equation to write cost and revenue as functions of the price, p. Then write profit as a function of price. (d) Graph profit against price. Find the price that maximizes profits. What is this profit? 3

When the price, p, charged for a boat tour was $25, the average number of passengers per week, N, was 500. When the price was reduced to $20, the average number of passengers per week increased to 650. Find a formula for the demand curve, assuming that it is linear. 3

Table 1.30 gives data for the linear demand curve for a product, where p is the price of the product and q is the quantity sold every month at that price. Find formulas for the following functions. Interpret their slopes in terms of demand. (a) q as a function of p. (b) p as a function of q. Table 1.30 p (dollars) 16 18 20 22 24 q (tons) 500 460 420 380 340 3

The demand curve for a product is given by q = 120,000 500p and the supply curve is given by q = 1000p for 0 q 120,000, where price is in dollars. (a) At a price of $100, what quantity are consumers willing to buy and what quantity are producers willing to supply? Will the market push prices up or down? (b) Find the equilibrium price and quantity. Does your answer to part (a) support the observation that market forces tend to push prices closer to the equilibrium price? 3

World production, Q, of zinc in thousands of metric tons and the value, P, in dollars per metric ton are given54 in Table 1.31. Plot the value as a function of production. Sketch a possible supply curve. 3

A taxi company has an annual budget of $720,000 to spend on drivers and car replacement. Drivers cost the company $30,000 each and car replacements cost $20,000 each. (a) What is the companys budget constraint equation? Let d be the number of drivers paid and c be the number of cars replaced. (b) Find and interpret both intercepts of the graph of the equation. 3

A company has a total budget of $500,000 and spends this budget on raw materials and personnel. The company uses m units of raw materials, at a cost of $100 per unit, and hires r employees, at a cost of $25,000 each. (a) What is the equation of the companys budget constraint? (b) Solve for m as a function of r. (c) Solve for r as a function of m. 3

You have a budget of $2000 for the year to cover your books and social outings. Books cost (on average) $80 each and social outings cost (on average) $20 each. Let b denote the number of books purchased per year and s denote the number of social outings in a year. (a) What is the equation of your budget constraint? (b) Graph the budget constraint. (It does not matter which variable you put on which axis.) (c) Find the vertical and horizontal intercepts, and give a financial interpretation for each. 3

Linear supply and demand curves are shown in Figure 1.59, with price on the vertical axis. (a) Label the equilibrium price p0 and the equilibrium quantity q0 on the axes. (b) Explain the effect on equilibrium price and quantity if the slope, p/q, of the supply curve increases. Illustrate your answer graphically. (c) Explain the effect on equilibrium price and quantity if the slope, p/q, of the demand curve becomes more negative. Illustrate your answer graphically. Supply Demand q p Figure 1.59 3

The demand for a product is given by p = 9010q. Find the ratio !!!! Relative change in demand Relative change in price !!!! if the price changes from p = 50 to p = 51. Interpret this ratio. 3

A demand curve has equation q = 100 5p, where p is price in dollars. A $2 tax is imposed on consumers. Find the equation of the new demand curve. Sketch both curves. 4

A supply curve has equation q = 4p 20, where p is price in dollars. A $2 tax is imposed on suppliers. Find the equation of the new supply curve. Sketch both curves. 4

A tax of $8 per unit is imposed on the supplier of an item. The original supply curve is q = 0.5p 25 and the demand curve is q = 1650.5p, where p is price in dollars. Find the equilibrium price and quantity before and after the tax is imposed. 4

The demand and supply curves for a product are given in terms of price, p, by q = 2500 20p and q = 10p 500. (a) Find the equilibrium price and quantity. Represent your answers on a graph. (b) A specific tax of $6 per unit is imposed on suppliers. Find the new equilibrium price and quantity. Represent your answers on the graph. (c) How much of the $6 tax is paid by consumers and how much by producers? (d) What is the total tax revenue received by the government? 4

In Example 8, the demand and supply curves are given by q = 100 2p and q = 3p 50, respectively; the equilibrium price is $30 and the equilibrium quantity is 40 units. A sales tax of 5% is imposed on the consumer. (a) Find the equation of the new demand and supply curves. (b) Find the new equilibrium price and quantity. (c) How much is paid in taxes on each unit? How much of this is paid by the consumer and how much by the producer? (d) How much tax does the government collect? 4

Answer the questions in Problem 43, assuming that the 5% sales tax is imposed on the supplier instead of the consumer.