Ch 2 6 Distance rate x time Objective

Ch 2. 6 Distance = rate x time Objective: To solve word problems involving distance, speed, and time.

Definitions Distance: Units include feet, yards, miles, meters, etc Rate/Speed: distance Units are in fraction format time ( ) Time: Units include seconds, minutes, hours, days, etc. Average Speed: The total distance traveled divided by the total time taken for a particular journey. Transitive Property: if a = b and b = c then a = c

Example of Unit Conversion The world record for the 100 yard dash is 9. 4 seconds. How fast is this in miles/hr? miles hr Equivalent units 100 yards = 9. 4 seconds 1 yard = 3 feet 60 seconds = 1 minute 5280 feet = 1 mile 60 minutes = 1 hour

Rules 1. Underline the numbers and units in the problems Same direction 2. THINK the problem through & DRAW a picture. Opposite direction Round trip 3. Input the information into the table below d = r t 4. Solve each line using d = r t convert units if necessary Note: distances can be added together. times can be added together. rates can NOT be added together! You must use d = r t

Example 1 Jane and her friends are driving to a cabin for the weekend. It is 150 miles away. If they drive 50 mi/hr, how long will it take them to get there? 50 miles 1 hr 100 miles 2 hrs 150 miles hrs 3? hrs

Example 1 Table method Jane and her friends are going to a cabin for the weekend. It is 150 miles away. If they are drive 50 mi/hr, how long will it take them to get there? Jane d = r 150 mi 50 mi/hr t t hr 150 mi = 50 mi t hr 50 mi hr 3 hr = t Solve for t

Example 1 Same Direction Jane and her friends are driving to a cabin for the weekend. It is 150 miles away. If they drive 50 mi/hr, how long will it take them to get there? 50 miles 1 hr 100 miles 2 hrs 150 miles 3 hrs Her friend leaves one hour later. She wants to arrive at the same time as Jane. How fast must she drive? ? miles 75 miles 1 hr 150 miles 2 hrs

Example 1 Table method Jane and her friends are going to a cabin for the weekend. It is 150 miles away. If they are drive 50 mi/hr, how long will it take them to get there? Her friend leaves one hour later. She wants to arrive at the same time as Jane. How fast must she drive? d = r 150 mi/hr Jane Same r Her friend 150 mi distance 150 mi = r (3 – 1)hr 2 hr 75 mi/hr = r t 3 3 - 1 Solve for r

Example 2 Same Direction A passenger train leaves the train depot 2 hours after a freight train left the same depot. The freight train is traveling 20 mph slower than the passenger train. Find the speed of the passenger train, if it overtakes the freight train in three hours. (r-20) mi 2(r-20) mi 3(r-20) mi 4(r-20) mi 5(r-20) mi 1 hr 2 hrs 3 hrs 5 hrs 4 hrs = r miles 1 hr 2 r mi 2 hrs 3 r mi 3 hrs

Example 2 Table method A passenger train leaves the train depot 2 hours after a freight train left the same depot. The freight train is traveling 20 mph slower than the passenger train. Find the speed of the passenger train, if it overtakes the freight train in three hours. Same Passenger distance Freight Passenger d=r 3 = 3 r d d d = r r r - 20 t 3 Solve for r 3+ 2 Freight d = (r – 20) (3 + 2) = 5 r - 100 -2 r = -100 r = 50 m/h

Classwork 1) An aircraft carrier left Azores and traveled toward Madagascar at an average speed of 12 mph. A container ship left one hour later and traveled in the same direction at an average speed of 15 mph. How long did the aircraft carrier travel before the container ship caught up? Same carrier distance container d d d = r 12 15 t t Solve for t t-1 carrier container d = 12 t d = 15(t – 1) 12 t = 15 t - 15 t = 5 hours

Classwork 2) Daniel left the mall and traveled toward the mountains at an average speed of 30 mph. Joe left one hour later and traveled in the same direction at an average speed of 40 mph. How long did Daniel travel before Joe caught up? Same Daniel distance Joe d d d = r 30 40 t t Solve for t t -1 Daniel Joe d = 30 t d = 40(t – 1) 30 t = 40 t - 40 t = 4 hours

Example 3 Opposite Direction Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet? 14 miles 7 miles 8 miles 16 miles 1 hr + ½ hr = 1. 5 hrs 45 miles

Example 3 Table method Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet? Same time Cyclist 1 Cyclist 2 Cyclist 1 d 1 = 14 t = 14 t d = d 1 45 - d 1 45 r 14 16 t = 1. 5 h t t t Solve for t Cyclist 2 45 – d 1 = 16 t – d 1 = 16 t - 45 d 1 = -16 t + 45

Example 4 Opposite Direction Maria left the White House at the same time as Trevon. They traveled in opposite directions. Trevon traveled at a speed of 39 mph. After two hours they were 140 miles apart. How fast did Maria drive? 78 miles 2 hrs r 1 hr 39 miles 1 hr 78 + 140 miles 2 r 2 r 2 hrs = 140

Example 4 Table method Maria left the White House at the same time as Trevon. They traveled in opposite directions. Trevon traveled at a speed of 39 mph. After two hours they were 140 miles apart. How fast did Maria drive? Same time Maria Trevon Maria d 1 = r 2 = 2 r d = d 1 140 - d 1 140 r r 39 r = 31 mph t 2 2 Solve for r Trevon 140 – d 1 = 39 2 – d 1 = 78 - 140 d 1 = 62

Classwork 3) An Air Force plane left Los Angeles and flew toward Jakarta at an average speed of 350 mph. A cargo plane left some time later flying in the opposite direction with an average speed of 260 mph. After the Air Force plane had flown for 11 hours the planes were 4630 miles apart. How long had the cargo plane been flying? d = Same Air Force d 1 distance cargo 4630 - d 1 4630 r 350 260 t 11 t Solve for t Air Force cargo d = 350(11) 4630 - d = 260 t 4630 - 3850 = 260 t t = 3 hours

Classwork 4) A diesel train left Bangalore and traveled west at an average speed of 85 mph. A freight train left two hours later and traveled in the opposite direction with an average speed of 35 mph. How many hours did the freight train travel before the trains were 890 miles apart? Same diesel distance freight diesel d = 85 t d = d 1 890 - d 1 890 r 85 35 t t t-2 Solve for t-2 freight 890 - d = 35(t - 2) 890 – 85 t = 35 t - 70 t = 8 hours t – 2 = 6 hours

Example 5 Round Trip A boat travels for three hours with a current of 3 mph and then returns the same distance against the current in four hours. What is the boat's speed in calm water? +3 with current +3 1 hr +3 2 hrs with current +3 r with current d = 3 r + 9 r with current +3 r 3 hrs +3

Example 5 Round Trip A boat travels for three hours with a current of 3 mph and then returns the same distance against the current in four hours. What is the boat's speed in calm water? − 3 miles 4 hrs − 3 miles 3 hrs 2 hrs − 3 miles 1 hr against current r− 3 with current d = 3 r + 9 r− 3 against current d = 4 r − 12

Example 5 Table method A boat travels for three hours with a current of 3 mph and then returns the same distance against the current in four hours. What is the boat's speed in calm water? d Same with current d distance against current d = r r+ 3 r-3 t 3 4 Solve for r with current against current d = (r + 3) 3 = 3 r + 9 d = (r – 3) 4 = 4 r - 12 9 = r - 12 r = 21 m/h

Classwork 5) Kathryn took a trip to City Hall and back. The trip there took two hours and the trip back took five hours. She averaged 36 mph faster on the trip there than on the return trip. What was Kathryn’s average speed on the trip there? Same there distance back there d d d = r r + 36 r back d = 5 r d = (r + 36) 2 2 r + 72 = 5 r r = 24 mph r + 36 = 60 mph t 2 5 Solve for r + 36

Classwork 6) An aircraft carrier traveled to Madagascar and back. It took one hour longer to go there than it did to come back. The average speed on the trip there was 20 mph. The average speed on the way back was 25 mph. How long did it take for the aircraft carrier to fly to Madagascar? d Same to Madagascar d distance back d = r 20 25 to Madagascar back d = 20(t + 1) d = 25 t 20 t + 20 = 25 t t = 4 hours t + 1 = 5 hours t t + 1 Solve for t+1 t