Resources

Resources

Solve equations with variables on both sides: word problems

16

. A cone has a surface area of

16 \pi \mathrm{m}^{2}

and a volume of

8 \pi \mathrm{m}^{3}

. If a similar cone has a surface area of

100 \pi \mathrm{m}^{2}

. What is the similarity ratio of the smaller cone to the larger cone? What is the volume of the larger cone?

Giant kelp is one of the fastest-growing organisms on the planet. Desmond told his little brother that kelp grows about

1.38 \times 10^{-2}

feet per minute. What would be the most appropriate unit for Desmond to use instead of feet per minute?

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(A) inches per minute

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(B) feet per second

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(C) feet per hour

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(D) feet per day

Standard to Slope Intercept Form

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1

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Put the following equation of a line into slope-intercept form, simplifying all fractions.

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12 x-20 y=-100

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1

2

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0

5

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7

10

2

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12

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The table shows the approximate number of ounces of chlorine that should be added to a pool based on the number of gallons it holds.

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Number of Gallons of Water

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x

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4

500

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7

000

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Number of Ounces of Chlorine

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y

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0

585

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0

91

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Find the unit rate for the ounces of chlorine to the gallons of water. Then write an equation to represent the situation.

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y= \square

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5

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7

10

2

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10

12

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10

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The table shows the approximate number of ounces of chlorine that should be added to a pool based on the number of gallons it holds.

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\begin{tabular}{|l|l|l|}

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\hline Number of Gallons of Water

(\boldsymbol{x})

4

500

7

000

\newline

\hline Number of Ounces of Chlorine

(\boldsymbol{y})

0

585

0

91

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Find the unit rate for the ounces of chlorine to the gallons of water. Then write an equation to represent the situation.

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\square

\square

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y=

\square

7

-42

X

Y

650

miles apart, Train \#

1

travels from City

X

Y

60

mile/hour. Train \#

2

travels from City

Y

X

70

mile/hour. Train \#

1

8

00

X

2

10

00

am from City Y. What time will the two trains meet?

150

words per minute on average in Hungarian, and

190

words per minute on average in Polish. She once gave cooking instructions in Hungarian, followed by cleaning instructions in Polish. Sakura spent

5

minutes total giving both instructions, and spoke

270

more words in Polish than in Hungarian.

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How long did Sakura speak in Hungarian, and how long did she speak in Polish?

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Sakura spoke for

\square

minutes in Hungarian and for

\square

minutes in Polish.

Vera studies the extinction of the bear population of Siberia over time. The following function gives the number of bears

t

years since Vera started tracking it:

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B(t)=2190 \cdot e^{-0.3 t}

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What is the instantaneous rate of change of the number of bears after

2

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1

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1202

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1202

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-361

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-361

A person stands

12

meters east of an intersection and watches a car driving away from the intersection to the north at

4

meters per second.

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At a certain instant, the car is

9

meters from the intersection.

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What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

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1

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2

4

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4 \sqrt{10}

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15

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\frac{20}{3}

Two cars are driving away from an intersection in perpendicular directions.

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The first car's velocity is

7

meters per second and the second car's velocity is

3

meters per second.

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At a certain instant, the first car is

5

meters from the intersection and the second car is

12

meters from the intersection.

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What is the rate of change of the distance between the cars at that instant (in meters per second)?

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1

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\frac{99}{13}

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\mathbf{1 3}

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\sqrt{58}

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\frac{71}{13}

Two cars are driving away from an intersection in perpendicular directions.

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The first car's velocity is

5

meters per second and the second car's velocity is

8

meters per second.

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At a certain instant, the first car is

15

meters from the intersection and the second car is

20

meters from the intersection.

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What is the rate of change of the distance between the cars at that instant (in meters per second)?

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1

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13

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25

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9

4

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\sqrt{89}

29

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

7

meters per minute.

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At a certain instant, the bottom of the ladder is

21

meters from the wall.

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What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

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1

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\frac{20}{3}

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20

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\frac{147}{20}

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7

20

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

8

meters per minute.

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At a certain instant, the top of the ladder is

12

meters from the ground.

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What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

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1

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\frac{32}{3}

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8

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24

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6

39

-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at

10

meters per minute.

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At a certain instant, the bottom of the ladder is

\mathbf{3 6}

meters from the wall.

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What is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)?

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1

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-\frac{25}{6}

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-12

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-\mathbf{1 0}

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-24

A person stands

12

meters east of an intersection and watches a car driving away from the intersection to the north at

4

meters per second.

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At a certain instant, the car is

9

meters from the intersection.

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What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

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1

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\frac{20}{3}

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15

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4 \sqrt{10}

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2

4

Two cars are driving away from an intersection in perpendicular directions.

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The first car's velocity is

7

meters per second and the second car's velocity is

3

meters per second.

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At a certain instant, the first car is

5

meters from the intersection and the second car is

12

meters from the intersection.

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What is the rate of change of the distance between the cars at that instant (in meters per second)?

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1

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\frac{99}{13}

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\frac{71}{13}

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\mathbf{1 3}

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\sqrt{58}

Two cars are driving away from an intersection in perpendicular directions.

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The first car's velocity is

5

meters per second and the second car's velocity is

8

meters per second.

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At a certain instant, the first car is

15

meters from the intersection and the second car is

20

meters from the intersection.

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What is the rate of change of the distance between the cars at that instant (in meters per second)?

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1

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13

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9

4

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25

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\sqrt{89}

25

-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at

3

5

meters per minute.

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At a certain instant, the top of the ladder is

7

meters from the ground.

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What is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)?

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1

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-3

5

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-12

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-7

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-24

29

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

7

meters per minute.

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At a certain instant, the bottom of the ladder is

21

meters from the wall.

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What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

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1

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7

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\frac{147}{20}

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20

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\frac{20}{3}

20

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

8

meters per minute.

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At a certain instant, the top of the ladder is

12

meters from the ground.

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What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

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1

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8

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6

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\frac{32}{3}

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24

A person stands

30

meters east of an intersection and watches a car driving away from the intersection to the north at

17

meters per second.

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At a certain instant, the car is

16

meters from the intersection.

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What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

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1

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\sqrt{1189}

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34

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36

125

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8

Khiloe Hoenig e Book Goog

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My OLDEST DAUGHILR IS M

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Assignment Player

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:- bigideasmath.com/MRL/public/app/\#/student/assessment;isPlayerWindow=true;assignmentld=

5419

9

-4016

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10

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M

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15

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MODELING REAL LIFE You are riding on a zip line. Your speed is

15

miles per hour. What is your speed in feet per second?

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\square

feet per second

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7

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8

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9

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10

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11

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12

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13

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14

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15

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16

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support powered b

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CalcChat and CaleV

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20

feet per second directly to a flowerbed from its hive. The bee stays at the flowerbed for

12

minutes, and then flies directly back to the hive at

12

feet per second. It is away from the hive for a total of

17

minutes. What equation can you use to find the distance of the flowerbed from the hive?

6

. At a given instant, a cyclist is at a point

A

along a straight path

A P B

. He cycles along

A P

at an average speed of

8 \mathrm{~m} / \mathrm{s}

P Q

at an average speed of

6 \mathrm{~m} / \mathrm{s}

, to reach a point

Q

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A B=90 \mathrm{~m}

B Q=40 \mathrm{~m}

, determine the distance between

A

A P B

0

such that he will be able to reach

Q

in the shortest time.

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Get Al Tutoring New e:

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^{2}

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4

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Google Classroom

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You might need: [igleculator

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What is the area of the region between

2

consecutive polnts where the graphs of

f(x)=\cos (x)

g(x)=-\cos (x)+2

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1

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4

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4 \pi

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2

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2 \pi

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Show Calculator

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Brport a problem

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s^{\prime} \quad

5

19

DeltaMath Student Application

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Desmos | Scientlific Calculator

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3345436

22863483

11367

6

689

6035

83

8

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ReadTheory | Readi...

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Space huggers - Op.

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1

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Solve the equation for all real solutions in simplest form.

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3 z^{2}-z+5=6

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Answer Attempt y out of

2

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9

) Additional Solution

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z=

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During a mayoral election, candidate A spent

\$960,000

on her campaign, and candidate B spent

\$720,000

on her campaign. What is the ratio of candidate A's campaign spending to candidate B's campaign spending?

Julia and Lena started reading a science fiction book at the same time. During the first week, Lena read

10

more pages than Julia. The following week, Lena read

14

pages, and Julia read the same number of pages she read the first week. Now they are on the same page.

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Which equation can you use to find

j

, the number of pages Julia read during the first week?

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j + 10 + 14 = j + j

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j + 10 = j + j + 14

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How many pages did Julia read during the first week?

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A plane and a helicopter are both flying toward Clover City Airport. The plane is

1,200

miles from the airport, and it is flying at a constant speed of

510

miles per hour. The helicopter is

500

miles from the airport, and it is flying at a constant speed of

160

miles per hour.

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Which equation can you use to find

h

, the number of hours it will take for the plane and the helicopter to be the same distance from the airport?

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1,200 - 500h = 510 - 160h

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1,200 - 510h = 500 - 160h

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How many hours will it take for the plane and the helicopter to be the same distance from the airport?

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Simplify any fractions.

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A plane and a helicopter are both flying toward Clover City Airport. The plane is

1,200

miles from the airport, and it is flying at a constant speed of

510

miles per hour. The helicopter is

500

miles from the airport, and it is flying at a constant speed of

160

miles per hour.

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Which equation can you use to find

h

, the number of hours it will take for the plane and the helicopter to be the same distance from the airport?

\newline

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1,200 - 510h = 500 - 160h

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1,200 - 500h = 510 - 160h

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How many hours will it take for the plane and the helicopter to be the same distance from the airport?

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Simplify any fractions.

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a period of time of

24

24

minutes is how many times longer than a period of time of

24

24

Today, the population of Canyon Falls is

22

500

and the population of Swift Creek is

15

200

. The population of Canyon Falls is decreasing at the rate of

740

people each year while the population of Swift Creek is increasing at the rate of

1

500

people each year. Assuming these rates continue into the future, in how many years from today will the population of Swift Creek equal twice the population of Canyon Falls?

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1

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9

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\mathbf{1 0}

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\mathbf{1 1}

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12

310

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5

-5

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Which pay is equivalent to a rate of

\$ 16

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\$ 48

3

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\$ 60

5

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\$ 120

12

Water and other materials necessary for biological activity in trees are absorbed by the roots and transported throughout the stem and branches in thin, hollow tubes in the xylem. Water may travel as fast as

150

feet per hour. Which of the following inequalities best describes the number of minutes,

m

, after being absorbed by the root, that it takes for water to reach a height of

250

feet above the ground?

The concession stand at a football game sells hot dogs and drinks. It costs

34

25

2

1

drink. It costs

\$ 7.00

3

2

Irinks. Which of the following systems of equations can be solved to letermine the cost,

h

, of each hot dog and the cost,

d

, of each drink?

A piece of glass with an initial temperature of

99^{\circ} \mathrm{C}

is cooled at a rate of

3.5^{\circ} \mathrm{C}

per minute. At the same time, a piece of copper with an initial temperature of

0^{\circ} \mathrm{C}

2.5^{\circ} \mathrm{C}

per minute. Which of the following systems of equations can be used to solve for the temperature,

T

, in degrees Celsius, and the time,

m

, in minutes, when the glass and copper reach the same temperatures?

\rightarrow \mathrm{C}

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khanacademy.org/math/cc-seventh-grade-math/cc

-7

th-fractions-decimals/cc

-7

th-rates/e/rate_problems_1?classCode=DVFGV.

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uncommonschools.org bookmarks

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(i) We've updated our Cookie Notice. Please review it now.

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c^{3}

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peyton.j.fleming

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7

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775

12 \frac{1}{2}

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How many steps did he average per minute?

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COURSE: TTH GRADE > UNIT

2

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1

: Rate problems with steps per minute fractions

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23

11

59

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Rates with fractions

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Stuck? Review related articles/videos or use a hint.

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Report a problem

At the beginning of the week, Paloma opened a new box of

275

coffee creamers. Sixty-four coffee creamers were used during the week. Which equation could be used to find

x

, the number of coffee creamers left at the end of the week?

18

bags of glass to the recycling center. He still has

6

bags of plastic to take to the recycling center. Which equation could be used to find

x

, the total number of bags of glass and plastic Silas will take to the recycling center?

In basketball, the number of points you score is given by

3x+2y+z

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x

is three-pointers,

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y

is two-pointers, and

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z

is free throws (onepointers). What is the total number of points scored by a player who makes

3

three-pointers,

5

two-pointers, and

6

Dan and Makiko started doing their homework at the same time. It took Dan twice as long to finish his homework than Makiko did. If Dan also took

40

minutes longer than Makiko did, which of the following systems of equations could be used to determine

d

, the amount of time Dan took to finish his homework in minutes, and

m

, the amount of time it took Makiko to finish her homework in minutes?

A stone fell from the top of a cliff into the ocean.

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In the air, it had an average speed of

16 \mathrm{~m} / \mathrm{s}

. In the water, it had an average speed of

3 \mathrm{~m} / \mathrm{s}

before hitting the seabed. The total distance from the top of the cliff to the seabed is

127

meters, and the stone's entire fall took

12

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How long did the stone fall in the air and how long did it fall in the water?

A commercial airplane that is

1,500

2,500

-mile journey is traveling at

450

knots in still air when it picks up a tailwind of

150

knots (in the same direction). If

\newline

h

is the number of hours remaining in the airplane's flight, which of the following equations best describes the situation?

\newline

1

= 1.15

miles per hour (mph)

\newline

1

\newline

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1,500 + 690h = 2,500

\newline

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1,500 + 600h = 2,500

\newline

\newline

1,500 - 600h = 2,500

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\newline

2,500

0

\$11.50

per hour as a barista and

\$40

per hour as a tutor. What is the ratio of Helen's hourly earning as a barista to her hourly earning as a tutor?

Anton must finish reading a

369

-page book in the next

7

days. Which of the following inequalities can be used to represent

\newline

x

, the average number of pages he must read each day to finish the book on or ahead of schedule?

A robot is expected to filter pollution out of at least

350

liters of air and water. It filters air at the rate of

50

liters per minute, and it filters water at the rate of

20

liters per minute. Which of the following inequalities represents the number of minutes the robot should filter air

(A)

(W)

to meet this expectation?

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1

\newline

50A+20W \geq 350

\newline

50A+20W \leq 350

\newline

20A+50W \geq 350

\newline

20A+50W \leq 350

A cart is slowing down at a rate of

2 t+60

centimeters per second per second (where

t

is the time in seconds).

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By how many centimeters per second does the cart slow down between

t=5

t=15

\newline

1

\newline

20

\newline

75

\newline

800

\newline

875