1 Lesson 4.2.3 Distance, Speed and Time Distance, Speed and Time.

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Lesson 4.2.3Lesson 4.2.3

Distance, Speed and Time

Distance, Speed and Time

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Lesson

4.2.3Distance, Speed and TimeDistance, Speed and Time

California Standard:Algebra and Functions 4.2Solve multistep problems involving rate, average speed, distance, and time or a direct variation.

What it means for you:You’ll learn the formula for speed, and how to use it to solve problems.

Key words:• speed• distance• time• formula

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Distance, Speed and TimeDistance, Speed and TimeLesson

4.2.3

Speed is a rate — it’s the distance you travel per unit of time.

There’s a formula that links speed, distance, and time — and you’re going to use it in this Lesson.

55 miles per hour is the speed limit on some roads. If you drive steadily at this speed, you’ll travel 55 miles every hour.

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Distance, Speed and TimeDistance, Speed and Time

Speed is a Rate

Lesson

4.2.3

Speed is a rate. It is the distance traveled in a certain amount of time.

Speed can be measured in lots of different units, such as miles per hour, meters per second, inches per minute...

The formula for speed is:distance

timespeed =

2 hours10 miles per hour

20 miles

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Example 1

Solution follows…

Lesson

4.2.3

Gila walked 6 miles in 8 hours. What was Gila’s average speed?

Solution

Use the formula, and substitute in the values from the question.

Gila’s average speed was 0.75 miles per hour.

distancetime

speed = =6 miles8 hours

= (6 ÷ 8) miles per hour = 0.75 miles per hour

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Rearrange the Equation to Find Other Unknowns

Lesson

4.2.3

You can rearrange the speed formula, and use it to find distance or time.

To change the equation into an equation that gives distance in terms of speed and time, multiply both sides of the equation by time.

distancetime

speed =

distance × timetimespeed × time = distance = speed × time

You can find the equation for time in terms of speed and distance in a similar way.

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Example 2

Solution follows…

Lesson

4.2.3

Alyssa runs for 0.5 hours at a speed of 11 kilometers per hour. How far does she run?

Solution

Use the formula for distance, and substitute the values for speed and time.

Distance = speed × time

= 11 kilometers per hour × 0.5 hours

= 5.5 kilometers

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Example 3

Solution follows…

Lesson

4.2.3

Andy is planning a walk. He walks at an average speed of 3 miles per hour, and plans to cover 15 miles. How long should his walk take him?

Solution

You need to rearrange the speed formula.

distance = speed × time

Divide both sides by speedspeed × time

speeddistancespeed

=

Solution continues…

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Example 3

Lesson

4.2.3

Andy is planning a walk. He walks at an average speed of 3 miles per hour, and plans to cover 15 miles. How long should his walk take him?

Solution (continued)

Now you can use the formula to answer the question:

Andy’s walk should take him 5 hours.

distancespeed

time =15 miles3 mph

= = (15 ÷ 3) hours = 5 hours

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Guided Practice

Solution follows…

Lesson

4.2.3

1. Juan ran in a marathon that was 26 miles long. If his time was 4 hours, what was his average speed?

2. Moesha goes to school every day by bike. The journey is 6 miles long, and takes her 0.6 hours. What is her average speed?

3. Monica travels 6 miles to work at a speed of 30 miles per hour. How long does the journey take her each morning?

Speed = distance ÷ time = 26 ÷ 4 = 6.5 miles per hour

Speed = distance ÷ time = 6 ÷ 0.6 = 10 miles per hour

Time = distance ÷ speed = 6 ÷ 30 = 0.2 hours or 12 minutes

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Guided Practice

Solution follows…

Lesson

4.2.3

Josh has been walking for 5 hours at a speed of 4 miles per hour.

4. His walk is 22 miles long. How far does he have left to walk?

5. How much longer will he take if he continues at the same speed?

Find how far Josh has already walked. Distance = speed × time = 4 × 5 = 20 milesSo Josh has 22 – 20 = 2 miles left to walk

Time = distance ÷ speed = 2 ÷ 4 = 0.5 hours or 30 minutes

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So her speed for the last hour = x miles per hour.


Example 4

Solution follows…

Lesson

4.2.3

On a three-hour bike ride, a cyclist rode 58 miles. The first two hours were downhill, so the cyclist rode 5 miles per hour quicker than she did for the last hour. a) What was her speed for the first two hours? b) What was her speed for the last hour?

You need to write an equation using the information given.

Let the cyclist’s speed for the first two hours be (x + 5) mi/h.

Solution


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Example 4

Lesson

4.2.3



Total distance = +distance traveled in first two hours

distance traveled in last hour

distance = speed × time

58 = + (x + 5) × 2 x × 1


58 = + 2x + 10 x

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a) The speed for the first two hours was (x + 5) = 16 + 5 = 21 mi/h

b) So the speed for the last hour was x = 16 mi/h


Example 4

Lesson

4.2.3



58 = 2x + 10 + x

58 = 3x + 10

Solve the equation to find x:

48 = 3x x = 16

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Guided Practice

Solution follows…

Lesson

4.2.3

6. Train A travels 20 mi/h faster than Train B. Train A takes 3 hours to go between two cities, and Train B takes 4 hours to travel the same distance.

How fast does each train travel?Let d = distance between the two citiesTrain A speed = d ÷ 3Train B speed = d ÷ 4

Train A speed = 240 ÷ 3 = 80 mi/h, Train B speed = 240 ÷ 4 = 60 mi/h

Train A speed = Train B speed + 20 mi/hd ÷ 3 = (d ÷ 4) + 204d = 3d + 240 d = 240 mi

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Independent Practice

Solution follows…

Lesson

4.2.3

1. A mouse ran at a speed of 3 meters per second for 30 seconds. How far did it travel in this time?

2. A slug crawls at 70 inches per hour. How long will it take it to crawl 630 inches?

3. A shark swims at 7 miles per hour for 2 hours, and then at 9 miles per hour for 3 hours. How far does it travel altogether?

90 meters

41 miles

9 hours

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Independent Practice

Solution follows…

Lesson

4.2.3

4. Bike J moves at a rate of x miles per hour for 2 hours. Bike K travels at 0.5x miles per hour for 4 hours. Which bike travels the furthest?

5. On a two-day journey, you travel 500 miles in total. On the first day you travel for 5 hours at an average speed of 60 mi/h. On the second day you travel for 4 hours. What’s your average speed for these 4 hours?

Both travel the same distance

50 mi/h

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Distance, Speed and TimeDistance, Speed and TimeLesson

4.2.3

Round UpRound Up

You need to remember the formula for speed.

If you know this, you can rearrange it to figure out the formulas for distance and time when you need them — so that’s two less things to remember.

distancetime

speed =

1 Lesson 4.2.3 Distance, Speed and Time Distance, Speed and Time.

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