"Section 1.7 Applications Interest, Mixture, Uniform Motion"
Section 1.7 Applications: Interest, Mixture, Uniform Motion, Constant-Rate Jobs 1. Review of steps in Setting Up Applied Problems (p. 142) The steps to follow in setting up an applied problem, given in earlier in Section 1.1, are given below: • Read the problem carefully, perhaps two or three times. Pay particular attention to the question being asked in order to identify what you are looking for. If you can, determine realistic possibilities for the answer. • Assign a letter (variable) to represent what you are looking for and, if necessary, express any remaining unknown quantities in terms of this variable. • Make a list of all the known facts, and translate them into mathematical expressions. These may take the form of an equation (or, later, an inequality) involving the variable. If possible, draw an appropriately labeled diagram to assist you. Sometimes a table or chart helps you organize your ideas. • Solve the equation for the variable, and then answer the question, usually using a complete sentence. • Check the answer with the facts in the problem. If it agrees, congratulations! If it does not agree, try again. 2. Interest Problems (p. 142) Use the simple interest formula to set up interest problems involving simple interest. If a principal of P dollars is borrowed for a period of t years at a per annnum interest rate r, expressed as a decimal, the interest I charged is: I = Prt. Example 1: A bank has $4,000,000 to loan to clients and must obtain an average return of 8% per year. If rates are to be 6% and 11%, how much can the bank loan at each rate and still obtain the 8% average rate of return? Note: Portions of this document are excerpted from College Algebra by Sullivan. 3. Mixture Problems (p. 144) If two quantities are mixed, set up a chart with three rows and three columns. One row is for each quantity in the mixture and for the mixture itself. The three columns are • The value or cost per unit of each quantity • The number of units of each quantity, and • Total cost or amount of each quantity. Using the appropriate entries from the chart and the information from the word problem set up an equation that represents the problem and solve the problem. Give your answer in English words. Example 2: A candy store sells boxes of candy containing caramels and crèmes. Each box sells for $12.50 and holds 30 pieces of candy (all pieces are the same size). If the caramels cost $0.25 to produce and the crèmes cost $0.45 to produce, how many of each should be in a box to make a profit of $3. Note: Portions of this document are excerpted from College Algebra by Sullivan. 4. Uniform Motion (p. 145) An object in uniform motion is moving at a constant speed. The equation that represents the relationship between the object’s average velocity, v, the distance s covered in time t is given by the formula: s = vt To solve uniform motion problems, set up a table with a row for each of the objects in motion or for each of the legs of a journey; and three columns: velocity, time and distance. Use the information in the word problem and the table to write an equation that represents the word problem. Solve the equation, and then state your answer in English words. Example 3: A motorboat heads upstream on a river that has a current of 3 miles per hour. The trip upstream takes 5 hours, and the return trip takes 2.5 hours. What is the speed of the motorboat? (Assume that the motorboat maintains a constant speed relative to the water). Note: Portions of this document are excerpted from College Algebra by Sullivan. 5. Constant Rate Jobs (p. 146) If a job can be completed in t units of time (i.e. hours, minutes, etc.), 1 then of the job can be completed in one unit of time. To solve t problems that involve working together to do a job, make a chart. Make a row for each of the workers contributing to the completion of the job, and make three columns, one for the number of time-units required for each worker to complete the job working alone, another for the part of the job completed by that worker in one time-unit, and one for the time that both workers work together. Example 4: A hot water pipe can fill a tub in 7 minutes and the cold water pipe can fill the tub in 4 minutes. How long does it take for both pipes working together to fill the tub? Round your answer to the nearest tenth of a minute. Section 1.7 Answers Example 1: $2,400,000 at 6% and $1,600,000 at 11% Example 2: 20 pieces of caramels and 10 pieces of crèmes Example 3: 9 miles per hour 6 Example 4: 2 minutes 11 Note: Portions of this document are excerpted from College Algebra by Sullivan.