Solved: Break-Even Analysis Suppose that the fixed cost

Problem 1E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. f(2)

Problem 2E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. f(4)

Problem 4E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. f(?1)

Problem 3E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. f(?3)

Problem 48A Education Cost The 2009–2010 budget for the California State University system projected a fixed cost of $486,000 at each of five off-campus centers, plus a marginal cost of $1140 per student. Source: California State University. a. Find a formula for the cost at each center, C(x), as a linear function of x, the number of students. ________________ b. The budget projected 500 students at each center. Calculate the total cost at each center. ________________ c. Suppose, due to budget cuts, that each center is limited to $1 million. What is the maximum number of students that each center can then support?

Problem 5E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. g(1.5)

Problem 6E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. g(2.5)

Problem 8E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. g (?3/4)

Problem 7E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. g (?1/2)

Problem 9E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. f(t)

Problem 10E For Exercise let f(x) = 7 ? 5x and g(x) = 2x ? 3. Find the following. g(k2)

Problem 11E In Exercise decide whether the statement is true or false. To find the x- intercept of the graph of a linear function, we solve y = f(x) = 0, and to find the y-intercept, we evaluate f(0).

Problem 13E In Exercise decide whether the statement is true or false. The slope of the graph of a linear function cannot be undefined.

Problem 15E Describe what fixed costs and marginal costs mean to a company.

Problem 14E In Exercise decide whether the statement is true or false. The graph of f(x) = ax is a straight line that passes through the origin.

Problem 16E In a few sentences, explain why the price of a commodity not already at its equilibrium price should move in that direction.

Problem 12E In Exercise decide whether the statement is true or false. The graph of f(x) = ?5 is a vertical line.

Problem 17E Explain why a linear function may not be adequate for describing the supply and demand functions.

Problem 18E In your own words, describe the break-even quantity, how to find it, and what it indicates.

Problem 19E Write a linear cost function for situation. Identify all variables used. A Lake Tahoe resort charges a snowboard rental fee of $10 plus $2.25 per hour.

Problem 20E Write a linear cost function for situation. Identify all variables used. An Internet site for downloading music charges a $10 registration fee plus 99 cents per downloaded song.

Problem 21E Write a linear cost function for situation. Identify all variables used. A parking garage charges 2 dollars plus 75 cents per half-hour.

Problem 22E Write a linear cost function for situation. Identify all variables used. For a one-day rental, a car rental firm charges $44 plus 28 cents per mile.

Problem 24E Assume that situation can be expressed as a linear cost function. Find the cost function in case. Fixed cost: $35; 8 items cost $395 to produce.

Problem 25E Assume that situation can be expressed as a linear cost function. Find the cost function in case. Marginal cost: $75; 50 items cost $4300 to produce.

Problem 23E Assume that situation can be expressed as a linear cost function. Find the cost function in case. Fixed cost: $100; 50 items cost $1600 to produce.

Problem 26E Assume that situation can be expressed as a linear cost function. Find the cost function in case. Marginal cost: $120; 700 items cost $96,500 to produce.

Problem 27A Supply and Demand Suppose that the demand and price for a certain model of a youth wristwatch are related by p = D(q) = 16 ? 1.25q, where p is the price (in dollars) and q is the quantity demanded (in hundreds). Find the price at each level of demand. a. 0 watches b. 400 watches c. 800 watches Find the quantity demanded for the watch at each price. d. $8 e. $10 f. $12 g. Graph p = 16 ? 1.25q. Suppose the price and supply of the watch are related by p = S(q) = 0.75q, where p is the price (in dollars) and q is the quantity supplied (in hundreds) of watches. Find the quantity supplied at each price. h. $0 i. $10 j. $20 k. Graph p = 0.75q on the same axis used for part g. l. Find the equilibrium quantity and the equilibrium price.

Problem 28A Supply and Demand Suppose that the demand and price for strawberries are related by p = D(q) = 5 ? 0.25q where p is the price (in dollars) and q is the quantity demanded (in hundreds of quarts). Find the price at each level of demand. a. 0 quarts b. 400 quarts c. 840 quarts Find the quantity demanded for the strawberries at each price. d. $4.50 e. $3.25 f. $2.40 g. Graph p = 5 ? 0.25q. Suppose the price and supply of strawberries are related by p = S(q) = 0.25q, where p is the price (in dollars) and q is the quantity supplied (in hundreds of quarts) of strawberries. Find the quantity supplied at each price. h. $0 i. $2 j. $4.50 k. Graph p = 0.75q on the same axis used for part g. l. Find the equilibrium quantity and the equilibrium price.

Problem 29A Supply and Demand Let the supply and demand functions for butter pecan ice cream be given by and where p is the price in dollars and q is the number of 10-gallon tubs. a. Graph these on the same axes. ________________ b. Find the equilibrium quantity and the equilibrium price. (Hint: The way to divide by a fraction is to multiply by its reciprocal.)

Problem 30A Supply and Demand Let the supply and demand functions for sugar be given by p = S(q) = 1.4q ? 0.6 and p = D(q) = ?2q + 3.2, where p is the price per pound and q is the quantity in thousands of pounds. a. Graph these on the same axes. ________________ b. Find the equilibrium quantity and the equilibrium price.

Problem 31A Supply and Demand Suppose that the supply function for honey is p = S(q) = 0.3q + 2.7, where p is the price in dollars for an 8-oz container and q is the quantity in barrels. Suppose also that the equilibrium price is $4.50 and the demand is 2 barrels when the price is $6.10. Find an equation for the demand function, assuming it is linear.

Problem 32A Supply and Demand Suppose that the supply function for walnuts is p = S(q) = 0.25q + 3.6, where p is the price in dollars per pound and q is the quantity in bushels. Suppose also that the equilibrium price is $5.85, and the demand is 4 bushels when the price is $7.60. Find an equation for the demand function, assuming it is linear.

Problem 35A Marginal Cost of Coffee The manager of a restaurant found that the cost to produce 100 cups of coffee is $11.02, while the cost to produce 400 cups is $40.12. Assume the cost C(x) is a linear function of x, the number of cups produced. a. Find a formula for C(x). ________________ b. What is the fixed cost? ________________ c. Find the total cost of producing 1000 cups. ________________ d. Find the total cost of producing 1001 cups. ________________ e. Find the marginal cost of the 1001st cup. ________________ f. What is the marginal cost of any cup and what does this mean to the manager?

Problem 33A T-Shirt Cost Joanne Wendelken sells silk-screened T-shirts at community festivals and crafts fairs. Her marginal cost to produce one T-shirt is $3.50. Her total cost to produce 60 T-shirts is $300, and she sells them for $9 each. a. Find the linear cost function for Joanne’s T-shirt production. ________________ b. How many T-shirts must she produce and sell in order to break even? ________________ c. How many T-shirts must she produce and sell to make a profit of $500?

Problem 39A Break-Even Analysis You are the manager of a firm. You are considering the manufacture of a new product, so you ask the accounting department for cost estimates and the sales department for sales estimates. After you receive the data, you must decide whether to go ahead with production of the new product. Analyze the data in Exercise (find a break-even quantity) and then decide what you would do in each case. Also write the profit function. C(x) = 85x + 900;R(x) = 105x; no more than 38 units can be sold.

Problem 36A Marginal Cost of a New Plant In deciding whether to set up a new manufacturing plant, company analysts have decided that a linear function is a reasonable estimation for the total cost C(x) in dollars to produce x items. They estimate the cost to produce 10,000 items as $547,500, and the cost to produce 50,000 items as $737,500. a. Find a formula for C(x). ________________ b. Find the fixed cost. ________________ c. Find the total cost to produce 100,000 items. ________________ d. Find the marginal cost of the items to be produced in this plant and what does this mean to the manager?

Problem 34A Publishing Costs Alfred Juarez owns a small publishing house specializing in Latin American poetry. His fixed cost to produce a typical poetry volume is $525, and his total cost to produce 1000 copies of the book is $2675. His books sell for $4.95 each. a. Find the linear cost function for Alfred’s book production. ________________ b. How many poetry books must he produce and sell in order to break even? ________________ c. How many books must he produce and sell to make a profit of $1000?

Problem 37A Break-Even Analysis Producing x units of tacos costs C(x) = 5x + 20; revenue is R(x) = 15x, where C(x) and R(x) are in dollars. a. What is the break-even quantity? ________________ b. What is the profit from 100 units? ________________ c. How many units will produce a profit of $500? Q

Problem 40A Break-Even Analysis You are the manager of a firm. You are considering the manufacture of a new product, so you ask the accounting department for cost estimates and the sales department for sales estimates. After you receive the data, you must decide whether to go ahead with production of the new product. Analyze the data in Exercise (find a break-even quantity) and then decide what you would do in each case. Also write the profit function. C(x) = 105x + 6000;R(x) = 250x; no more than 400 units can be sold.

Problem 38A Break-Even Analysis To produce x units of a religious medal costs C(x) = 12x + 39. The revenue is R(x) = 25x. Both C(x) and R(x) are in dollars. a. Find the break-even quantity. ________________ b. Find the profit from 250 units. ________________ c. Find the number of units that must be produced for a profit of $130.

Problem 41A Break-Even Analysis You are the manager of a firm. You are considering the manufacture of a new product, so you ask the accounting department for cost estimates and the sales department for sales estimates. After you receive the data, you must decide whether to go ahead with production of the new product. Analyze the data in Exercise (find a break-even quantity) and then decide what you would do in each case. Also write the profit function. C(x) = 70x + 500;R(x) = 60x (Hint: What does a negative break-even quantity mean?)

Problem 44A Break-Even Analysis Suppose that the fixed cost for a product is $650 and the break-even quantity is 25. Find the marginal profit (the slope of the linear profit function).

Problem 43A Break-Even Analysis Suppose that the fixed cost for a product is $400 and the break-even quantity is 80. Find the marginal profit (the slope of the linear profit function).

Problem 46A Body Temperature You may have heard that the average temperature of the human body is 98.6°. Recent experiments show that the actual figure is closer to 98.2°. The figure of 98.6 comes from experiments done by Carl Wunderlich in 1868. But Wunderlich measured the temperatures in degrees Celsius and rounded the average to the nearest degree, giving 37°C as the average temperature. Source: Science News. a. What is the Fahrenheit equivalent of 37°C? ________________ b. Given that Wunderlich rounded to the nearest degree Celsius, his experiments tell us that the actual average human body temperature is somewhere between 36.5°C and 37.5°C. Find what this range corresponds to in degrees Fahrenheit.

Problem 47A Temperature Find the temperature at which the Celsius and Fahrenheit temperatures are numerically equal.

Problem 45A Temperature Use the formula for conversion between Fahrenheit and Celsius derived in Example to convert each temperature. a. 58°F to Celsius b. ? 20°F to Celsius c. 50°C to Fahrenheit EXAMPLE Temperature Derive an equation relating F and C. SOLUTION To derive the required linear equation, first find the slope using the given ordered pairs, (0, 32) and (100, 212). The F-intercept of the graph is 32, so by the slope-intercept form, the equation of the line is With simple algebra this equation can be rewritten to give C in terms of F:

Problem 42A Break-Even Analysis You are the manager of a firm. You are considering the manufacture of a new product, so you ask the accounting department for cost estimates and the sales department for sales estimates. After you receive the data, you must decide whether to go ahead with production of the new product. Analyze the data in Exercise (find a break-even quantity) and then decide what you would do in each case. Also write the profit function. C(x) = 1000x + 5000;R(x) = 900x