10.7 HW Answers

1.

2.

3.

4.

x =

−b± b2 −4ac2a

−3,no solution

73, two solutions

32, two solutions

5.

6.

7.

8.

0,one solution

−100,no solution

D

A

9.

10.

11.

12.

one solution

−11,no solution

20, two x −intercepts

0,one x −intercept

13.

14.

15.

16.

−15,no x−intercept

one solution

no solution

two solutions

17.

18.

19.

x =1±2 2

no solution

x −4x+3( ) x−2( )

Scoring Your Homework• Count how many problems you missed

or didn’t do

• 0-1 missed = 10• 2-3 missed = 9• 4-5 missed = 8• 6-7 missed = 7• 8-9 missed = 6

• 10-11 missed = 5• 12-13 missed = 4• 14-15 missed = 3• 16-17 missed = 2• 18-19 missed = 1• 20-21 missed = 0

Rate Problems

A rate is way of expressing how fast something occurs. For example, how long it takes to walk or run one mile, how far a car goes in one hour, how long a pump takes to fill a swimming pool, how long it takes a worker to paint a house, etc. Rates are usually expressed as some kind of event per unit of time: mi/hr (rate of speed), $/hr (rate of earning), cubic feet/sec (rate of pumping water)

Let's look closer at the speed of a car. If the car is travelling at 30 miles each hour, we say its rate or speed is 30mi/hr. If it travels constantly at this speed for 10 hours, how far will it go? Clearly the time must be multiplied to the speed to get the distance travelled. Therefore:

distance = rate time d r t

We can use this formula to solve different kinds of problems involving rates. Use a chart to organize your information.

Ex 1: During a 456 mile trip, Lucy drove the first two hours at an average speed of 48 mi/hr. During the remainder of the trip, her friend Ethyl drove for another 8 hrs. What was Ethyl's average speed?

Strategy to solve:1) Write known amounts in table2) Calculate any blanks possible3) Use a variable for unknown amount4) Write an equation using using two known quantities and only one variable

d rt

distance = rate time

Lucy

Ethyl

Ex 1: During a 456 mile trip, Lucy drove the first two hours at an average speed of 48 mi/hr. During the remainder of the trip, her friend Ethyl drove for another 8 hrs. What was Ethyl's average speed?

456

248 96

8r 360

distance = rate time

Lucy

Ethyl

456

248 96

8r 360

8r = 360

r = 45 mph

distance = rate time

b) During a 840 mile flight, a small plane averages a speed of 160 mi/hr for the first 3 hours when one engine fails. For the remaining 3 hours of the flight, its speed was reduced to what average speed?

840

3160 480

3r 360

1st

2nd

3r = 360

r = 120mph

distance = rate time

840

3160 480

3r 360

1st

2nd

Ex 2: Train A leaves the train station travelling at an average speed of 40 mi/hr. Eight hours later, Train B leaves in the same direction as Train A, but is travelling at an average speed of 60 mi/hr. How long will it be before Train B catches up to Train A?

In this table, your variable needs to represent the desiredquantity, which is how long Train B travels until it catches up to Train A. For Train A's time, it has been travelling 8 hours longer than Train B, so represent this fact using an expression with "t". You will have two unknown distances, but what do you know about the two distances that each train will be from the station at the point where Train B catches up to Train A? Make your equation using this fact.

distance = rate time

Train A

Train B

Ex 2: Train A leaves the train station travelling at an average speed of 40 mi/hr. Eight hours later, Train B leaves in the same direction as Train A, but is travelling at an average speed of 60 mi/hr. How long will it be before Train B catches up to Train A?

t40 d

t – 860 d

distance = rate time

Train A

Train B

t40 d

t – 860 d

60(t – 8) = 40t

60t – 480 = 40t20t – 480 = 0

20t = 480t = 24 hrs

Try:a) Rex Racer starts off around a go-cart track and is averaging a speed of 20 ft/s. His friend Sparky, starts 5 seconds later and averages 25 ft/s around the track. How long will it be before Sparky catches up to Rex?

distance = rate time

t20 d

t – 525 d

Rex

Sparky

distance = rate time

t20 d

t – 525 d

Rex

Sparky

25(t – 5) = 20t

25t – 125 = 20t5t – 125 = 0

5t = 125t = 25 sec

Work ProblemsAnother type of rate is that at which work is done. Usually work is expressed as how much is done per unit of time:Cars painted per day, items assembled per hour, clay pots made per hour, lawns mowed per hour, bolts torqued per minute, etc. The total amount of work done clearly depends on how fast you work (rate) and how much time is spent on the task. Work done can be considered as 1 item done if it is something such as: one car painted, one house painted, one swimming pool filled by a pump, etc.

= rate timeWork W r t

Ex 3: With spraying equipment, John can paint the wood trim on a small house in 8 hours. His assistant, Bart, must paint by hand since there is only one sprayer, and he needs 12 hours to complete the same type of job. If they work together on the same house, how long should it take them to complete the job?

Since each worker will only do part of the job, their work is a fraction of the one job that will be completed. The work that they do then, must add up to one job.

Work = rate time

1 house

t18

t 112

John

Bart

1

8t

1

12t

1 11

8 12t t

1 1 1

8 12t t

24

3 2 24t t

24

5 24t

24

24

5t

44

5 hrs

Ex 4: A large water pump can fill a standard size swimming pool in 4 hours, while medium size water pump will take 6 hours to fill the same pool. Working both pumps at once, how long will it take to fill 3 standard size pools?

Work = rate time

3 pools

t14

t16

large

medium

1

4t

1

6t

1 1 3

4 6t t

12

3 2 36t t

12

5 36t

12

36

5t

17

5 hrs

Try:c) Cletus can split a cord of wood in 4 days. His friend Gomer can split a cord in 2 days. How long would it take to split a cord of wood if they work together?

Work = rate time

1 cord

t14

t12

Cletus

Gomer

1

4t

1

2t

1 1 1

4 2t t

4

2 4t t

4

3 4t

4

4

3t

11

3 days

d) Henry and Aaron own an oak wall-unit business. Henry can stain their large wall-unit in 3 hours and Aaron takes 4 hours. How long would it take them to stain 2 wall units if they work together?

Work = rate time

2 walls

t13

t14

Henry

Aaron

1

3t

1

4t

1 1 2

3 4t t

12

4 3 24t t

12

7 24t

12

24

7t

33

7 hrs

HW #1

1. Andy's average speed driving on a 4 hour trip was 45 mi/hr. During the first 3 hours he drove 40 mi/hr. What was his average speed for the last hour of the trip?

distance = rate time

4 45 =

340 120

1r 60

1st

2nd

180

distance = rate time

4 45 =

340 120

1r 60

1st

2nd

180

r = 60 mph