car, in years 0 (new car)

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1 Chapter 2.4: Applications of Linear Equations In this section, we discuss applications of linear equations how we can use linear equations to model situations in our lives. We already saw some examples in the previous sections, such as adding the same amount of money to a coin jar each week, and this table which shows car depreciation: Age of the Price car, in years 0 (new car) This table has the equation yy = 1300xx Remember, to be linear, the same amount must be added (or subtracted) each time that is, the slope must be constant. EXAMPLES a. We begin with a new example about car depreciation, but this time worded slightly differently than the example above. Brianna buys a car in 2010 for $15,000. By 2015, the price she could get if she sold the car has decreased to $7,000. Assume the relationship between the year and the price of the car is linear. Find the slope of this line and explain what it means about the car s value, then find the equation of the line. (Let 2010 be year zero.) Even though the numbers in this problem are not written in table form, we could still do so, with age in years as one variable and money as the other: Age, in years Price Notice that we had to figure out that the car is 5 years old in Alternatively, we could write the information as two points: (0,15000) and (5,7000). Either way we chose to write the information, the slope is found using the formula cchaaaaaaaa iiii oooooooooooo mm = cchaaaaaaaa iiii iiiiiiiiii = yy 2 yy = = 8000 xx 2 xx = 1600 As expected, we have a negative slope, since the car is decreasing in value. What does this mean about the value of the car? It means the price (or value) of the car is dropping by $1,600 each year. We know the slope tells us price per year because price is our y cchaaaaaaaa iiii oooooooooooo cchaaaaaaaa iiii pppppppppp (output) value and age in years is our x (input). mm = =. cchaaaaaaaa iiii iiiiiiiiii cchaaaaaaaa iiii aaaaaa Now we can find the equation of the line. We know yy = 1600xx + bb by substituting the slope we found yy = 1600xx since the y-intercept is (0,15000)

If you didn t realize that the y-intercept is given to you, you might try to solve for b using one of the points. That s okay, you will still get the same result!

+8000 +8000 15000 = b yy = 1600xx + 15000 We can now use the equation of the line above to find the value of the car in other years.

2 If you didn t realize that the y-intercept is given to you, you might try to solve for b using one of the points. That s okay, you will still get the same result! 7000 = 1600(5) + bb Use the point (5, 7000) = bb Solve to find b = b yy = 1600xx We can now use the equation of the line above to find the value of the car in other years. For example, in 2013, the car is 3 years old and we get: yy = 1,600(3) + 15,000 = $10,200. So, after three years, the car would be worth $10,200. b. At a local gym, you can decide to pay one of two options: Option 1, $100 a month OR Option 2, pay $60 a month and $7 for each time you go. If you tend to go to the gym 4 times per month, which option would you choose? Why? If you usually go to the gym 8 times per month, which option would you choose? Why? After you have thought about each option and answered the above questions, write an equation representing each option, with y representing the cost, and x representing the amount of times you go per month. If you go 4 times a month, the cost for option 1 is $100. The cost for option 2 is 60+7(4) = = $88, so option 2 is cheaper. If you go 8 times a month, the cost for option 1 is $100. The cost for option 2 is 60+7(8) = = $116, so option 2 is cheaper. To write the equation of the line for each option, look for a pattern in what you did for each option, above. For option 1, the answer was always 100, no matter how many times you went. Thus, the equation for option 1 is yy = 100. For option 2, you always added 60 + $7 number of times you go. This gives the equation yy = xx which you can rewrite if you like as yy = 7xx Now that you have the two equations, you can explain what the slope and y-intercept of each means within the context of the problem.

3 Interpretations: yy = 100 is a vertical line at 100, with 0 slope and y-intercept (0,100) The slope if 0 means there is no increase in cost each time you go. The y-intercept is (0,100), where the line hits the x-axis. This represents the cost of the membership if you go 0 times, which would still be $100. yy = 7xx + 60 has a slope of 7 and y-intercept of (0,60). The positive slope means you have an increase of $7 each time you go to the gym. The y-intercept of (0,60) means that if you go to the gym zero times, it still costs $60. This is also known as the fixed cost. c. The population of a town is 235,000. Due to its popular schools, the population increases by 1,200 people per year. Let P be the population, and t represent time in years. Write the linear equation, then use it to predict the population in 7 years. Next, use the equation to find out how many years it will take for the population to reach 257,800. Since we are starting at 235,000 and adding 1,200 each year, we get the equation PP = 235, ,200tt. If you like, you can rewrite this as PP = 1,200tt + 235,000 (but you do not have to). Another way to tell that the slope of the equation is 1,200 people per year, is the fact that this is the amount that the population changes by each year. After 7 years, you have added on 1,200 new people 7 times, so you have PP = 235, ,200(7) = 243,400 people. To find how many years it will take for the population to reach 257,800 is a little trickier. The number 257,800 represents the population, P. PP = 235, ,200tt 257,800 = 235, ,200tt Substitute 257,800 in for P. 235, ,000 Solve to find t. 228,000 = 1,200 tt / 1200 / = t It will take 19 years. The amount that our total changes by each time is the slope. d. Joel sells cars. He is paid $1,200 per week, plus a 5% commission on his sales. Let s represents dollar amount in sales, and E represent earnings per week. Write a linear equation representing his total earnings, then use the equation to find out how much money he needs to make in sales to earn $4,000 per week. To figure out the commission, we multiply 5%, or 0.05, times his sales. Since Joel starts at $1,200 and adds on 5% of any sales he makes, the equation is E = 1, s If he wants to make $4000 per week in earnings, that s E, so we have to solve to find s. EE = 1, ss 4,000 = 1, ss Substitute 4,000 in for E. 1,200 1,200 Solve to find s. 2,800 = 0.05ss / 0.05 / ,000 = s He will need to sell $56,000 worth of cars.

4 e. The relationship between the amount of times a snowy tree-cricket chirps per minute, and the temperature is linearly correlated. It the temperature is 60 degrees, the cricket chirps 85 times per minute. If the temperature is 80 degrees, the cricket chirps 205 times per minute. Find the equation of this line. Let T represent temperature, and let C represent chirps per minute. Interpret the slope. We can use the information, above to create two points each with temperature and cricket chirps: (60, 85) and (80, 205). Next, we use the points to find the slope: mm = = = 6. CC = 6TT + bb Place the slope in for m. Notice that we use C and T instead of y and x in our y = mx + b form. 85 = 6(60) + bb Place the slope in for m. 85 = bb Solve to find b = bb The equation is CC = 6TT 275 We interpret the slope, m = 6, to mean that for every 6 chirps, there is 1 degree increase in temperature. We know the slope is chirps per degree because when we calculated the slope, we subtracted: (205 85) cchiiiiiiii (80 60) dddddddddddddd. Since our slope is cchiiiiiiii dddddddddddddd, we interpret the slope as chirps per degree. I am a snowy tree-cricket and I really chirp this way! Notice that the intercept, (0, -275) does not make sense in this context. The intercept is saying that when it is zero degrees, the cricket chirps negative 275 times! In fact, a cricket does not chirp when it is that cold outside. The equation is only valid within a normal spring and summer temperature range. We can now use the equation to predict the temperature if the cricket chirps 140 times per minute. CC = 6TT = 6TT 275 Substitute 140 in for C Solve to find T. 410 = 6TT / 6 / = T It will be about We can also use the equation to predict how many chirps you should hear per minute when the temperature is 50 degrees.

CC = 6TT 275 CC = 6(50) 275 Substitute 50 in for T. CC = 300 275 = 24 There will be 25 chirps. f. Finally, we look at a linear cost and revenue model which can be used to find profit.

5 CC = 6TT 275 CC = 6(50) 275 Substitute 50 in for T. CC = = 24 There will be 25 chirps. f. Finally, we look at a linear cost and revenue model which can be used to find profit. Suppose the owner of a hotdog cart spends $1000 on equipment (the cart and cookware, etc.), plus $0.20 per hot dog. The amount he spends on the equipment is called the fixed cost, because that is how much he has to spend no matter how many hots dogs he gets. The cost function is written CC = 0.20xx , where x is the number of hotdogs. If the cart owner sells hotdogs for $1.50, his revenue is given by RR = 1.50xx. Notice that the revenue equation has an intercept at (0,0) if he sells 0 hotdogs, he makes $0. What is the equation for profit? Profit is always given by Revenue Cost, that is, the amount of money you take in from your business, minus the amount you spend on it. PP = RR CC = 1.50xx (0.20xx ). PP = 1.50xx 0.20xx 1000 PP = 1.30xx 1000 Distribute the negative. Simplify. How much profit will he have if he sells 200 hot dogs? What about 1000 hot dogs? If he sells 200 hot dogs, his profit will be PP = 1.30(200) 1000 = 740 If he sells 2000 hot dogs, his profit will be PP = 1.30(1000) 1000 = 300 Now, he makes $300. A negative profit means he is losing money.

b) According to the statistics above the graph, the slope is What are the units and meaning of this value?

b) According to the statistics above the graph, the slope is What are the units and meaning of this value? ! Name: Date: Hr: LINEAR MODELS Writing Motion Equations 1) Answer the following questions using the position vs. time graph of a runner in a race shown below. Be sure to show all work (formula, substitution,