U4L2I1.doc
Unit 4 – Day 3: Inverse VariationName: ____________________________INVESTIGATIONMany people take long car trips for business, vacations, and sometimes just commuting to work. While driving at slower speeds can save gas, driving at faster speeds can save time. For example, a 300-mile trip takes 6 hours at 50 mph, but only 5 hours at 60 mph. Think about how driving time would change if an average speed of 50 mph decreased to 40 mph.Suppose that your family is planning a 250 mile trip by car to visit relatives. Your average speed could vary from as little as 20 mph to 60 mph or more. Your average speed depends on what roads you take, traffic, weather, speed limits, and the driver’s preferred pace.1. How long will that 250 mile trip take if you average:20 mph40 mph60 mph2. Write a rule that gives time of this trip t as a function of the average driving speed s.a. Use your calculator to make a table showing data (speed, time) for the 250 mile trip. Then sketch the graph of this relation below.Speed(mph)101520253035404550556065Time(hours)330835406400Describe as accurately as possible, the pattern relating average speed and time of your trip. How is that pattern shown in the table and graph? CHECKPOINTIn Common Core Math 1, you learned about direct variation. In direct variation, two variables have a constant ratio so that if one variable increases, so does the other. 2668270457835Our road trip scenario is an example of inverse variation. Inverse variation is the opposite of direct variation in that values of the two variables change in an opposite manner – as one increases, the other decreases.Notice the shape of the graph of inverse variation. If the value of?x?_________________, then?y?____________________.?????If?x?______________________, the?y?value __________________.? We say that?y?varies inversely as the value of?x.?An?inverse variation?between 2 variables,?y?and?x, is a relationship that is expressed as: __________________________ where the variable?k?is called the?_____________________ _________ _________________________. GUIDED PRACTICE1 a. What equation will relate a set distance d, average speed s, and driving time t for a trip?b. How does an increase in average speed change the expected driving time for a fixed distance?c. How is your answer to part b shown in the graphs of (speed, time) relations for any fixed distance?2. The distance between New York and Los Angeles is approximately 3000 miles. a. How long will a trip from NY to LA takeBy airplane, averaging 450 mphBy car, averaging 60 mphBy bicycle, averaging 15 mphb. What equation gives the time t for the trip as a function of the average speed s?c. Make a table showing the relation between speed and time for speeds of 50 to500 mph. Make the graph on your calculator.Speed50100150200250300350400450500Timed. Which change in speed causes the greater change in time for the trip: an increase from 50 to 100 mph or an increase from 450 to 500 mph?INDEPENDENT PRACTICE1. R varies inversely with variable T. If R is 168 when T = 24, find R when T = 30. HINT: Remember y=kx2. The volume, V, of a gas varies inversely as the pressure, p, in a container. If the volume of a gas is 200cc when the pressure is 1.6 liters per square centimeter, find the volume (to the nearest tenth) when the pressure is 2.8 liters per sq centimeter.3. In science, one theory of life expectancy states that the lifespan of mammals varies inversely to the number of heartbeats per minute of the animal. If a gerbil's heart beats 360 times per minute and lives an average of 3.5 years, what would be the life expectancy of a human with an average of 72 beats per minute? Does this theory appear to hold for humans?4. The values (9.7, 8) and (3, y) are from an inverse variation. Find the missing value and round to the nearest hundredth.5. A drama club is planning a bus trip to New York City to see a Broadway play. The cost per person for the bus rental varies inversely as the number of people going on the trip. It will cost $30 per person if 44 people go on the trip. How much will it cost per person if 60 people go on the trip? Round your answer to the nearest cent, if necessary. ................
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