Problem Solving Using Systems

Transcription

1 Name Period Date: Topic: 3-6 Problem Solving Using Systems Essential Question Suppose you run a business that makes two products Each product requires a different amount of a raw material and sells for a different price, but you have a fixed amount of the raw material, and you want to receive a precise amount for the sale of your entire inventory How could you determine the amount of each product to produce in order to use all your raw material and achieve the desired amount for the sale of your products? Standard: A-REI6 Objective: Solve systems of linear equations exactly and approximately (eg, with graphs), focusing on pairs of linear equations in two variables To use systems of equations to solve problems Sometimes you can solve a problem involving two unknown quantities most easily by using a system of equations The five-step problem solving plan discussed in an earlier lesson is useful here However, you must set up and solve a system of equations rather than a single equation In this lesson, you will learn how to solve problems using a system of equations by studying examples Be sure you carefully explain the examples and completely work all the exercises Example 1: To use a certain computer data base, the charge is $30/h during the day and $1050/h at night If a research company paid $411 for 28 h of use, find the number of hours charged at the daytime rate and at the nighttime rate Summary

2 The problem asks for the number of hours charged at each rate Let d = the number of hours of use at the daytime rate Let n = the number of hours of use at the nighttime rate Set up a system of equations Total number of hours charged is 28 Total amount charged is 411 Solve the system Express d in terms of n in the first equation and substitute in the second equation Since, and, the solution is (6, 22) Check there were 6 h charged at the daytime rate and 22 h at the nighttime rate 2

3 Exercise 1: Kelly asked a bank teller to cash a $390 check using $20 bills and $50 bills If the teller gave her a total of 15 bills, how many of each type of bill did she receive? 3

4 Vocabulary: The following terms are used in connection with aircraft flight air speed wind speed tail wind head wind ground speed the speed of an aircraft in still air the speed of the wind a wind blowing in the same direction as the path of the aircraft a wind blowing in the direction opposite to the path of the aircraft the speed of the aircraft relative to the ground This vocabulary is commonly used in problems involving flight The following example illustrates this Example 2: To measure the speed of the jet stream (a high-speed, highaltitude, west-to-east wind), a weather-service plane flew 1800 km with the jet stream as a tail wind and then back again The eastbound flight took 2 h, and the westbound return flight took 3 h 20 min Find the speed of the jet stream and the air speed of the plane The problem asks for the plane's air speed and the speed of the wind Let p = the air speed in km/h Let w = the wind speed in km/h Using the fact that rate time = distance, construct a table for the plane s ground speed Eastbound 2 Westbound rate time = distance 4

5 From the right-hand column of the table, the plane flew a total distance east of and a total distance west of These distances are the same, and they are equal to 1800 km Therefore, we have the following two equations: Solve the system Using the linear-combination method, Check the air speed of the plane is 720 km/h and the speed of the jet stream is 180 km/h 5

6 Exercise 2: With a tail wind, a helicopter flew 300 mi in 1 h 40 min The return trip against the same wind took 20 min longer Find the wind speed and the air speed of the helicopter Using the fact that rate time = distance, construct a table for the helicopter s ground speed Outbound Return rate time = distance 6

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8 Example 3: Davis Rent-A-Car charges a fixed amount per weekly rental plus a charge for each mile driven A one-week trip of 520 miles cost $250, and a two-week trip of 800 miles cost $440 Find the weekly charge and the charge for each mile driven The problem asks for the weekly charge and the charge for each mile driven Let Let Therefore, the cost of renting a car is,, where x is the number of weeks, and y is the number of miles For the one-week trip of 520 miles For the two-week trip of 800 miles Solve the following system of equations: Multiply the first equation by 2 and add it to the second equation 8

9 Substituting into the first, original equation,, or Check Davis Rent-A-Car charges $120 per week plus $025 per mile driven Exercise 3: A deep sea charter boat company charges fishermen an hourly fee plus a charge for each large fish caught A six-hour fishing trip resulting in 7 large fish caught cost the fishermen $925, and a four-hour fishing trip resulting in 19 large fish caught cost the fishermen $975 What is the hourly charge for chartering the fishing boat, and what is the charge for each large fish caught? 9

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11 Example 4: If a particle starting with an initial speed of v 0 has a constant acceleration a, then its speed after t seconds is given by Find v 0 and a if when, and when The problem asks for the values of the initial speed and the constant acceleration Let v 0 = the initial speed Let a = the constant acceleration Then, and We ll solve this system using the substitution method Solving the first equation for v 0 gives, Substituting this expression for v 0 into the second equation gives, Solving for a gives, Substituting for a in the equation for v 0 gives, 11

12 and Check Exercise 4: You may have seen a video of an astronaut dropping a hammer and a feather in the airless environment of the moon This was intended to demonstrate that objects fall with the same acceleration in the absence of air-resistance You may have also noticed that the hammer fell much more slowly than it would have if it were dropped on the earth This is because the moon has a smaller gravity than the earth does Suppose a satellite orbiting the moon in a close orbit (close to the lunar surface) ejected a probe that struck the moon s surface This probe was ejected in such a way that it had no horizontal velocity At the instant the probe left the satellite was traveling directly down with a velocity of v 0 If the moon s gravitational acceleration is g m, then the probe s downward speed is given by the following equation: where t is the time after the probe s release Moreover, 15 s after its release, the probe was observed to be traveling at 47 m/s and 90 s after its release, the probe was traveling at 167 m/s What is the moon s gravitational constant (g m ), and what was the downward speed of the probe when it was released? 12

13 Check Class work: Homework: none p 132 Problems: 2-10 even P 133 Problems: 11-14, 16 13