THE “REALLY TOUGH STUFF” STORY PROBLEMS II THE “REALLY TOUGH STUFF”

General Directions for d = rt problems Read the problem carefully. Ask yourself, “What am I trying to find.” Determine what kind of a problem it is. Write your equation and solve Ask yourself, “Did I answer the question asked?”

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? What am I trying to find? How long does it take the MOTORBOAT to catch the canoe. In other words, how long is the motorboat on the water?

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? This problem contains speed (rate) and time.

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Use d = rt Make a table of the information.

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe Motorboat

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe 12 Motorboat

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe 12 Motorboat 21

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe 12 t Motorboat 21

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe 12 t Motorboat 21 t - 2

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe 12 t 12t Motorboat 21 t - 2

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? Rate Time Distance Canoe 12 t 12t Motorboat 21 t - 2 21(t - 2)

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? The motorboat has caught the canoe (same direction) The distances traveled by both must be the same. Set the two distances equal to each other and solve.

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2)

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2) 12𝑡=21𝑡−42

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2) 12𝑡=21𝑡−42 −9𝑡=−42

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2) 12𝑡=21𝑡−42 −9𝑡=−42 𝑡= −42 −9

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2) 12𝑡=21𝑡−42 −9𝑡=−42 𝑡= −42 −9 =4. 6

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2) You can convert your answer to hours and minutes if you like. 12𝑡=21𝑡−42 −9𝑡=−42 𝑡= −42 −9 =4. 6

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2 You can convert your answer to hours and minutes if you like. 12𝑡=21𝑡−42 The time you have found is the time for the canoe. You were not asked to find that. Find the motorboat’s time. −9𝑡=−42 𝑡= −42 −9 =4. 6

A canoe leaves a campsite and travels at an average speed of 12 mph A canoe leaves a campsite and travels at an average speed of 12 mph. Two hours later, a motorboat leaves the campsite and travels an average rate of 21 mph. After how many hours does the motorboat catch the canoe? 12𝑡=21(𝑡−2 You can convert your answer to hours and minutes if you like. 12𝑡=21𝑡−42 The time you have found is the time for the canoe. You were not asked to find that. Find the motorboat’s time. −9𝑡=−42 𝑡= −42 −9 =4. 6 It takes the motorboat 2 hours and 40 minutes to catch the canoe. Or you can just leave it 2. 6 hours.

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? This problem contains rate and time (but it is given as “clock” time.) If you go to a location, then come back, you have a “round trip” situation In a round trip, the distances are equal.

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Make a table. Add a column for the “clock” time.

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up Down

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 Down

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 Down 6

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 t Down 6

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 t Down 6 3 - t

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 t 4t Down 6 3 - t

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 t 4t Down 6 3 - t 6(3 – t)

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 t 4t 9:00 am Down 6 3 - t 6(3 – t)

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? Rate Time Distance Clock Up 4 t 4t 9:00 am Down 6 3 - t 6(3 – t) ????

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡)

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡) 4𝑡=18−6𝑡

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡) 4𝑡=18−6𝑡 10𝑡=18

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡) 4𝑡=18−6𝑡 10𝑡=18 𝑡= 18 10

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡) 4𝑡=18−6𝑡 10𝑡=18 𝑡= 18 10 =1.8 hours

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡) This does NOT answer the question asked. This is how long it takes to get to the top of the hill, not AT WHAT TIME? 4𝑡=18−6𝑡 10𝑡=18 𝑡= 18 10 =1.8 hours

Suppose you begin to hike up a hill at 9:00 a. m Suppose you begin to hike up a hill at 9:00 a.m. at an average rate of 4 km/hr. You hike back down the hill at an average rate of 6 km/hr. If you arrive back at your campsite at 12:00 p.m., at what time did you arrive at the top of the hill? 4𝑡=6(3−𝑡) This does NOT answer the question asked. This is how long it takes to get to the top of the hill, not AT WHAT TIME? 4𝑡=18−6𝑡 10𝑡=18 𝑡= 18 10 =1.4 hours If you left at 9:00 am, and it took 1.8 hours (or 1 hour 48 minutes), then you arrived at the top of the hill at 10:48 am.

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. The jets are flying in “opposite directions.” To solve an opposite direction problem, ADD the distances traveled to find the distance apart.

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Make a table.

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound Eastbound

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 Eastbound

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 Eastbound r

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 Eastbound r

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 Eastbound r

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r+25) Eastbound r

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r+25) Eastbound r 2r

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r+25) Eastbound r 2r 2𝑟+2 𝑟+25 =1250

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r+25) Eastbound r 2r 2𝑟+2 𝑟+25 =1250 2𝑟+2𝑟+50=1250

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r+25) Eastbound r 2r 2𝑟+2 𝑟+25 =1250 2𝑟+2𝑟+50=1250 4𝑟+50=1250

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r+25) Eastbound r 2r 2𝑟+2 𝑟+25 =1250 4𝑟=1200 2𝑟+2𝑟+50=1250 4𝑟+50=1250

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r + 25) Eastbound r 2r 2𝑟+2 𝑟+25 =1250 4𝑟=1200 2𝑟+2𝑟+50=1250 𝑟=300 4𝑟+50=1250

Two jets leave Dallas at the same time and fly in opposite directions Two jets leave Dallas at the same time and fly in opposite directions. One is flying west 25 mi/hr faster than the other. After 2 hours, the jets are 1250 miles apart. Find the speed of each jet. Rate Time Distance Westbound r + 25 2 2(r +25) Eastbound r 2r 2𝑟+2 𝑟+25 =1250 4𝑟=1200 2𝑟+2𝑟+50=1250 𝑟=300 4𝑟+50=1250 The rate (r) is the rate of the eastbound jet. Therefore, the westbound jet is flying at 325 mph.

Wrap-up There are three type of 𝑑=𝑟𝑡 problems. Same direction Round-trip Opposite direction In same direction and round-trip problems, you set the distances equal to each other. In opposite direction, you add the distances together.

Assignment 2.5B: 10 - 24