Advanced Algebra
Chapter 3 Answers
Solving Systems of Linear Equations and Inequalities
With the continuing growth of the Internet, more and more Internet Service Providers (ISPs) are springing into existence. If there's a service you're interested in, you may be able to decide which ISP offers a better deal by solving a system of equations.
Part A, Solving Systems of Equations Graphically
1. Find the fees charged by two ISPs that provide Internet access -- Sonic.net and dock.net.
a. Write an equation for the total cost of each service. Let y = the total cost and let x = the number of months. You may ignore JetLink's 2 month minimum. [Zap: y = 17.50x + 20; JetLink: y = 15x + 35]
b. Graph the two equations.
c. When will the cost of service from Zap be less than the cost of the service from JetLink? [Before the sixth month]
d. When will the cost of service from JetLink be less than the cost of the service from Zap? [After the sixth month]
e. When will the cost of the two services be the same? [During the sixth month]
Part B, Solving Systems of Equations Symbolically
2. Some Internet providers give their customers a choice between paying more for more access and paying less for measured usage. Find the fees for dialup services charged by Idiom Consulting.
a. Write an equation that expresses the total cost c of an "Unlimited" account for the first month of use for h hours of use in excess of 240 hours. Include the setup charge and the monthly charge but no additional fees. [c = 0.35 h + 40]
b. Write an equation that expresses the total cost c of an "Full" account for the first month of use for h hours of use in excess of 240 hours. Include the setup charge and the monthly charge but no additional fees. [c = 0.25 h + 60]
c. Solve the system of equations you have written. [h = 200, c = 110]
d. During the first month, how many hours in excess of 240 hours would you have to use Idiom's dialup modem service in order for a "Full" account to be cheaper than an "Unlimited" account. [More than 200 hours]
e. Suppose you decided to pay for one extra connection. Write new equations for the total cost c of the first month of "Unlimited" and "Full" accounts for h hours of use. Assume you use the service for less than 240 hours. ["Unlimited": c = 40 + h; "Full": c = 60 + 0.7h]
f. During the first month, with one extra connection and less than 240 hours of use, how many hours would you have to use Idiom's dialup modem service in order for a "Full" account to be cheaper than an "Unlimited" account? [More than 66 2/3 hours]
Part E, Making Connections
3. Telephone companies generally also give two options for service -- flat rate and measured rate. Find out one company's rate for flat rate and measured rate services.
a. Assume that measured-rate calls are charged at the rate of 5¢ per minute. What is the maximum amount of time that a measured-rate service can be used in one month without exceeding the cost of flat-rate service? (Remember that measured rate provides $3 worth of calls at no charge.) [ ($3.00 / $0.05) + ({$11.25 $6.00} / $0.05) = 60 + 105 = 165 minutes]
b. Find the answer to 3a, assuming that measured-rate calls are charged at the rate of 15¢ per minute. [ ($3.00 / $0.15) + ($11.25 {$6.00 / $0.15}) = 20 + 35 = 55 minutes]
c. Find the answer to 3a, assuming that measured-rate calls are charged at the rate of 25¢ per minute. [ ($3.00 / $0.25) + ($11.25 {$6.00 / $0.25}) = 12 + 21 = 33 minutes]
d. Are you more likely or less likely to use the measured rate service as the price per minute rises? Explain. [Less likely. When the cost per minute of the measured-rate service rises, making calls with this service becomes more costly.]
Systems of linear inequalities can be used to determine which combinations of products will fulfill a specific set of needs.
Part A, Systems of Inequalities
1. Suppose that, to be sure you are getting the Recommended Daily Allowance (RDA) of Vitamins A and C, you make a mango and watermelon salad.
a. What percentage of the RDA of Vitamins A and C does 1 serving of mangoes contain? [40% of vitamin A, 15% of vitamin C]
b. What percentage of the RDA of Vitamins A and C does 1 serving of watermelon contain? [20% of vitamin A, 25% of vitamin C]
c. In order for the salad to supply sufficient Vitamin A, it must contain at least 100% of the RDA of Vitamin A. Write an inequality expressing this condition, assuming the salad contains m servings of mangoes and w servings of watermelon . [ 0.4m + 0.2w ≥ 1]
d. Write an inequality stating that the salad contains at least 100% of the RDA of Vitamin C. [0.15m + 0.25w ≥ 1]
e. Graph your system of inequalities.
f. Give three examples of the numbers of servings of each fruit that will satisfy the requirements. [Answers will vary. Using whole number values at or near the vertices: 0 mangoes, 5 watermelon; 1 mango, 4 watermelon; 7 mangoes, 0 watermelon ]
Part C, Solving Linear Programming Problems
2. Like humans, animals have daily nutritional requirements.
a. Look in Table II to find the estimated daily amounts of calcium and potassium that dairy cattle need to maintain their health. [calcium, 116 g; potassium, 184 g]
b. A dairy farmer feeds his cows two types of mineral supplements per day.
Type Cost/kg Amount Calcium Amount Potassium A $0.90 200 g per kg of supplement 115 g per kg of supplement B $1.00 100 g per kg of supplement 230 g per kg of supplement
Let x represent the number of kilograms of Type A supplement a cow receives daily and let y represent the number of kilograms of Type B supplement a cow receives daily. Write an inequality giving the number of kilograms of calcium each cow receives daily, assuming the cow receives at least the "estimated daily amount" (Exercise 2a). Write an inequality giving the number of kilograms of potassium each cow receives daily. [0.2x + 0.1 y ≥ 0.116; 0.115x +0.115y ≥ 0.184]c. Write inequalities stating that each cow is to receive a combined total of no less than 1 kg of supplements per day and no more than 2 kg. [x + y ≥ 1; x + y ≤ 2]
d. Graph the inequalities, showing the feasible region and vertex points.
e. What is the daily per-cow cost function? [Cost = 0.9x + y]
f. Find the optimal solution, the least amount the farmer can spend daily per cow while still meeting the cow's nutritional needs. [Vertex point (0.4, 0.6) gives the minimum cost of $0.96 daily. Each cow receives 0.4 kg Type A and 0.6 kg Type B, supplying 140 g calcium and 184 g potassium.]
