Ch 4: Solving & Graphing Inequalities G) Distance = rate x time

Objective:

To solve word problems involving distance, speed, and time.

Distance:

Units include feet, yards, miles, meters, etc

Rate/Speed:

Units are in fraction format

Definitions

Time:

Units include seconds, minutes, hours, days, etc.

distance time( )

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 ConversionThe world record for the 100 yard dashis 9.4 seconds. How fast is this in miles/hr?

€

100 yds.9.4 sec .

Equivalent units

€

3 ft.1 yd.

€

1 mi.5280 ft.

€

60 sec.1 min .

€

60 min .1 hr.

€

×

€

×

€

×

€

×

€

=

€

=100(3)(1)(60)(60)9.4(1)(5280)(1)(1)

€

≈ 21.76 miles / hour

1 yard = 3 feet

5280 feet = 1 mile

60 seconds = 1 minute

60 minutes = 1 hour

100 yards = 9.4 seconds

mileshr

Rules1. Underline the numbers and units in the problems

2. THINK the problem through & DRAW a picture

3. Input the information into the table below

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

Same directionOpposite direction

Round trip

d = r t

Example 1Jane 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?

? hrs50 miles

1 hr100 miles

2 hrs 3 hrs150 miles

Example 1Jane 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?

150 mi 50mi/hr t Solve for t

150 mi = t hr50 mi

hr50 mi

50 mi hr

3 hr = t

d = r tJane

Table method

Example 1Jane 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?

Her friend leaves one hour later. She wants to arrive at the same time as Jane. How fast must she drive?

50 miles1 hr

100 miles2 hrs

2 hrs? miles

3 hrs150 miles

150 miles75 miles1 hr

Same Direction

Example 1

150 mi 50 mi/hr 33150 mi

Same distance

150 mi = (3 – 1)hrr

75 mi/hr = r

2 hr2 hr

d = r t

r Solve for r

Her friend leaves one hour later. She wants to arrive at the same time as Jane. How fast must she drive?

Jane Her friend

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?

- 1

Example 2

1 hr 2 hrs 3 hrs

3(r-20) mi

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.

4 hrs4(r-20) mi

5 hrs5(r-20) mi

3 hrs3r mi

2(r-20) mi(r-20) mi

2 hrs2r mi

1 hrr miles

=

Same Direction

Example 2A 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.

Passenger Freight

d 33d

d = r tr Solve for rSame

distance r + 2- 20

d = r 3Passenger Freight

d = (r – 20) (3 + 2) = 3r = 5r - 1003r

-2r = -100 r = 50 m/h

Table method

1)

Classwork

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?

d = r t

5 hours

12dd

Same distance 15 - 1

tt

Solve for tcarrier

container

d = 12tcarrier container

d = 15(t – 1)12t = 15t - 15

t =

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?

d = r t

4 hours

30dd

Same distance 40 - 1

tt

Solve for tDaniel

Joe

d = 30tDaniel Joe

d = 40(t – 1)30t = 40t - 40

t =

Classwork

Example 3

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?

45 miles

14 miles 16 miles

1 hr

7 miles 8 miles

+ ½ hr = 1.5 hrs

Opposite Direction

Example 3

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?

Cyclist 1 Cyclist 2

d1 tt45 - d1

d = r t14 Solve for tSame

time 16

d1 = 14 tCyclist 1 Cyclist 2

45 – d1 = 16 t = 14t = 16t - 45

45

– d1 = -16t + 45 d1

14t= -16t + 45t = 1.5 h

Table method

Example 4

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?

140 miles

2rr1 hr

39 miles1 hr2 hrs

78 miles2 hrs

78 + 2r = 140

Opposite Direction

Example 4

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?

Maria

Trevon

d1 22140 - d1

d = r tr Solve for rSame

time 39

d1 = r 2Maria Trevon

140 – d1 = 39 2 = 2r = 78 - 140

140

– d1

= 62 d1

2r= 62r = 31 mph

Table method

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 = r t

3 hours

350d1

4630 - d1

Same distance 260

11t

Solve for tAir Force

cargo

d = 350(11)Air Force cargo

4630 - d = 260t4630 - 3850 = 260t

4630

t =

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?

d = r t

t = 8 hours

85d1

890 - d1

Same distance 35

tt - 2

Solve for t - 2

diesel

freight

d = 85tdiesel freight

890 - d = 35(t - 2)890 – 85t = 35t - 70

890

6 hourst – 2 =

Classwork

Example 5

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?

1 hr 2 hrs 3 hrswith current

+ 3

with current with current

+ 3 + 3

r +3 r +3 r +3

d = 3r + 9with current

Round Trip

Example 5

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 hrs 2 hrs 1 hragainst current

− 3 miles

against current against current

− 3 miles − 3 miles

4 hrsagainst current

− 3 miles

d = 3r + 9 d = 4r − 12with current against current

r − 3r − 3r − 3r − 3

Round Trip

Example 5

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?

with current against current

d 34d

d = r tr Solve for rSame

distance r 3

d = (r + 3) 3with current against current

d = (r – 3) 4 = 3r + 9 = 4r - 123r + 9

9 = r - 12 r = 21 m/h

3+ -

Table method

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?

d = r t

60 mph

r + 36dd

Same distance r

25

Solve for r + 36

there

back

d = (r + 36) 2there back

d = 5r2r + 72 = 5r

r = 24 mph

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 = r t

5 hours

20dd

Same distance 25

t + 1t

Solve for t + 1

to Madagascar

back

d = 20(t + 1)to Madagascar back

d = 25t20t + 20 = 25t

t = 4 hours

t + 1 =

Classwork