Resources

Resources

Interpreting Linear Expressions

The scatterplot shows the temperatures,

y

, in degrees Celsius (

\text{C}^\circ

), of a cup of coffee cooling in a

22^\circ\text{C}

5

minute intervals. If

x

represents the time, in minutes, since the coffee had a temperature of

92^\circ\text{C}

, which of the following could be used to predict the temperature of the coffee for

0 \leq x \leq 60

Teresa currently has

\$1,000

in her bank account. The expression above models the amount of money in her account in

x

months if she deposits

\$250

into the account each month. Based on the expression, what is the amount, in dollars, Teresa will have in her account

6

months from now?

500

1

02

)^t The given equation models the amount of money in dollars, in a savings account years after the initial deposit was made. If the interest rate remains constant, which of the following equations models the amount of money in the savings account decades after the initial deposit was made?

500

1

02

)^t The given equation models the amount of money in dollars, in a savings account years after the initial deposit was made. If the interest rate remains constant, which of the following equations models the amount of money in the savings account decades after the initial deposit was made?

Paul opened a bakery. The net value of the bakery (in thousands of dollars)

t

months after its creation is modeled by Paul wants to know what his bakery's lowest net value will be

v(t)=2t^2-12t-141)

. Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation.

v(t)=

.What is the bakeries lowest net value?

Before he went to camp last summer, Mason's grandmother made him a patchwork quilt out of his old T-shirts. The quilt has an area of

q

square feet. To create the patchwork design, she used

12

rectangular patches, each with an area of

2

\newline

Which diagram models the story?

\newline

Which equation models the story?

\newline

\newline

\frac{q}{12} = 2

\newline

2q = 12

v=14.2(12.5-s)

\newline

v

, in meters per second, of an airplane

s

seconds after landing is given by the equation. What is the best interpretation of

14

2

as shown in the equation?

\newline

1

\newline

(A) The plane takes

14

2

seconds to stop.

\newline

(B) The position of the plane changes

14

2

meters every second.

\newline

(c) The velocity changes by

14

2

meters per second every second.

\newline

(D) The velocity of the plane just as it lands is

14

2

meters per second.

11

bookmarks at home. She bought

b

more bookmarks at the school book fair. Write an expression that shows the number of bookmarks Patty has now.

h(t)=-16.1 t^{2}+393 t+6

\newline

The function models

h

, the height of a firework shell in feet

t

seconds after launch. Which of the following statements is the best interpretation of the ordered pair

(10,2326)

An awning that covers a sliding glass door that is

88

inches tall forms an angle of

50

degrees with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the sun is more than

X = 55

degrees. See the figure. Find the length

L

An investor invested two equal sums of money for

1

year. The first sum was invested at the rate of

R\%

annual rate of interest and the interest compounding was half-yearly. The second sum was invested at the rate of

r\%

annual rate of interest and the interest compounding was quarterly. If at the end of the year, the amounts he received are equal, what is the value of

R

r

Noa drove from the Dead Sea up to Jerusalem. When she arrived in Jerusalem after

1.5

hours of driving, her altitude was

710

meters above sea level. Her altitude increased at a constant rate of

740

meters per hour.

\newline

y

represent Noa's altitude (in meters) relative to sea level after

x

\newline

Complete the equation for the relationship between the altitude and number of hours.

The function models

h(t) = -16.1t^2 + 393t + 6

, the height of a firework shell in feet

t

seconds after launch. Which of the following statements is the best interpretation of the ordered pair

(10,2326)

Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning. After

14

hours of burning, a candle has a height of

22

centimeters. After

33

hours of burning, its height is

12

5

centimeters. What is the height of the candle after

32

An investor invested two equal sums of money for

1

year. The first sum was invested at the rate of

R\%

annual rate of interest and the interest compounding was half-yearly. The second sum was invested at the rate of

r\%

annual rate of interest and the interest compounding was quarterly. If at the end of the year, the amounts he received are equal, what is the value of

R

r

Bethany sells roses and petunias. The expression

3r+2.5p

gives the cost, in dollars, of

r

p

petunias. What is the cost, in dollars, of

7

8

\newline

P(q)=-0.01(q-250)(q-80)

\newline

The given equation gives the profit,

P(q)

, in dollars, earned by a cupcake bakery when

q

cupcakes are produced. What is the best interpretation of the number

80

in this context?

\newline

1

\newline

80

is a number of cupcakes for which the profit is equal to

\$ 0

\newline

80

is the number of cupcakes that corresponds to the maximum profit.

\newline

80

is the number of cupcakes that corresponds to the minimum profit.

\newline

\$ 80

is the maximum profit, in dollars.

A pool is being filled. The following function gives the pool's water level (in centimeters) after

t

\newline

W(t)=4 \sqrt{t}+\ln (t+1)

\newline

What is the instantaneous rate of change of the pool's water level after

100

\newline

1

\newline

0

21

\newline

0

21

centimeters per second

\newline

44

6

\newline

44

6

centimeters per second

A pool is being filled. The following function gives the pool's water level (in centimeters) after

t

\newline

W(t)=4 \sqrt{t}+\ln (t+1)

\newline

What is the instantaneous rate of change of the pool's water level after

100

\newline

1

\newline

0

21

\newline

0

21

centimeters per second

\newline

44

6

\newline

44

6

centimeters per second

P

gives the population of a town

t

years after it was founded.

\newline

What is the best interpretation for the following statement?

\newline

The slope of the line tangent to the graph of

P

t=2

-10

\newline

1

\newline

2

years, the rate of change of the population decreased at a rate of

10

people per year per year.

\newline

2

years, the town's population decreased at a rate of

10

people per year.

\newline

(C) Over the first two years, the town's population decreased by

10

people per year.

\newline

2

years, the town's population decreased at a rate of

10

A weight that is attached to the end of a spring is pulled and then released. The function

H

gives its height, in centimeters, after

t

\newline

What is the best interpretation for the following statement?

\newline

H^{\prime}(0)=3

\newline

1

\newline

(A) When the weight is released, its height is

3

\newline

(B) When the weight is released, its mass is increasing at a rate of

3

grams per second.

\newline

(C) When the weight is released, its height is increasing at a rate of

3

\newline

(D) When the weight is released, its height is increasing at a rate of

3

centimeters per second.

P

gives the lion population at a certain game reserve

t

years after the reserve was established.

\newline

What is the best interpretation for the following statement?

\newline

The value of the derivative of

P

t=6

12

\newline

1

\newline

6

years, the lion population grows at a rate of

12

\newline

6

years, the lion population grows at a rate of

12

lions per year.

\newline

(C) The lion population grows at a rate of

12

6

\newline

6

years, there were

12

lions in the reserve.

A tank is being filled with a liquid. The function

V

gives the volume of liquid in the tank, in liters, after

t

\newline

What is the best interpretation for the following statement?

\newline

The value of the derivative of

V

t=1

2

\newline

1

\newline

1

minute, the tank was being filled at a rate of

2

\newline

1

minute, the tank had

2

liters of liquid.

\newline

1

minute, the tank was being filled at a rate of

2

liters per minute.

\newline

(D) During the first minute, the tank was being filled at a rate of

2

liters per minute.

Due to the tide, the water level rises and falls daily in Buzzard's Bay.

D

gives the depth of the water, in meters,

t

hours after midnight on a certain day.

\newline

What is the best interpretation for the following statement?

\newline

The value of the derivative of

D

t=8

-0

5

\newline

1

\newline

8

a.m. the water level decreased at a rate of

0

5

meters per hour.

\newline

8

a.m. the water level decreased at a rate of

0

5

\newline

8

a.m. the water level was

0

5

meters below sea level.

\newline

8

a.m. the water level decreased at a rate of

0

5

meters per hour.

Water is filling a pool. The function

W

gives the pool's water level (in meters) after

t

\newline

What is the best interpretation for the following statement?

\newline

The slope of the line tangent to the graph of

W

t=5

0

1

\newline

1

\newline

5

minutes, the pool is being filled at a rate of

0

1

meters per minute.

\newline

(B) The pool is being filled at a rate of

\frac{0.1}{5}

meters per minute.

\newline

5

minutes, the pool's water level is

0

1

\newline

5

minutes, the pool is being filled at a rate of

0

1

minutes per meter.

C

gives the costs (in dollars) of producing

x

liters of a certain sauce.

\newline

What is the best interpretation for the following statement?

\newline

The slope of the line tangent to the graph of

C

x=12

4

\newline

1

\newline

(A) At the time when

12

liters of sauce are produced, the costs increase by

4

\newline

(B) When the amount of sauce produced is

\mathbf{1 2}

liters, the costs increase at a rate of

4

dollars per liter.

\newline

(C) The costs of producing the sauce increase at a rate of

4

12

\newline

4

dollars to produce

12

liters of the sauce.

The side of the base of a square prism is decreasing at a rate of

7

kilometers per minute and the height of the prism is increasing at a rate of

10

kilometers per minute.

\newline

At a certain instant, the base's side is

4

kilometers and the height is

9

\newline

What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)?

\newline

1

\newline

-204

\newline

-148

\newline

148

\newline

204

\newline

The surface area of a square prism with base side

s

h

2 s^{2}+4 s h

The base of a triangle is decreasing at a rate of

13

millimeters per minute and the height of the triangle is increasing at a rate of

6

millimeters per minute.

\newline

At a certain instant, the base is

5

millimeters and the height is

1

\newline

What is the rate of change of the area of the triangle at that instant (in square millimeters per minute)?

\newline

1

\newline

-8

5

\newline

8

5

\newline

-21

5

\newline

21

5

10

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

3

meters per minute.

\newline

At a certain instant, the bottom of the ladder is

6

meters from the wall.

\newline

What is the rate of change of the area formed by the ladder at that instant (in square meters per minute)?

\newline

1

\newline

-\frac{7}{2}

\newline

7

\newline

12

\newline

-6

The radius of a sphere is decreasing at a rate of

4

centimeters per second.

\newline

At a certain instant, the radius is

10

\newline

What is the rate of change of the surface area of the sphere at that instant (in square centimeters per second)?

\newline

1

\newline

-64 \pi

\newline

-160 \pi

\newline

-320 \pi

\newline

-400 \pi

\newline

The surface area of a sphere with radius

r

4 \pi r^{2}

The side of the base of a square prism is decreasing at a rate of

7

kilometers per minute and the height of the prism is increasing at a rate of

10

kilometers per minute.

\newline

At a certain instant, the base's side is

4

kilometers and the height is

9

\newline

What is the rate of change of the surface area of the prism at that instant (in square kilometers per minute)?

\newline

1

\newline

148

\newline

-204

\newline

-148

\newline

204

\newline

The surface area of a square prism with base side

s

h

2 s^{2}+4 s h

A weight is attached to the end of a spring. Its height after

t

seconds is given by the equation

\newline

h(t)=5-2 \sin \left(\frac{2 \pi(t+1)}{7}\right) .

\newline

When does the weight first reach its maximum height? Give an exact answer.

\newline

t= \square

The present value

(P V)

of an investment is the amount that should be invested today at a specified interest rate in order to earn a certain amount at a future date. The amount desired is called the future value. For a future value of

\$ 10,000

, which of the following functions models the present value,

P V

, to be invested in a savings account earning

5 \%

interest compounded annually for

t

\newline

1

\newline

P V(t)=10,000(1.05)^{t}

\newline

P V(t)=10,000(1.05)^{-t}

\newline

P V(t)=10,000(1+0.05 t)

\newline

P V(t)=10,000(1-0.05 t)

C=0.85(130) h+15 T

\newline

The cost to rent a

90

-horsepower pontoon on Lake Pleasant for

h

hours, along with the rental of

T

water toys, during the Early Bird Special discount period, is given by the equation. What is the meaning of the

15 T

in this equation?

\newline

1

\newline

15 T

is the cost of the pontoon rental for

15

hours, with no water toys.

\newline

15 T

is the amount of the Early Bird Special discount.

\newline

15 T

is the cost of renting one water toy.

\newline

15 T

is the total cost of renting water toys.

m=0.15(8.6-s)

\newline

The amount in moles,

m

, of a reactant

s

seconds after the start of a chemical reaction is given by the equation. Which of the following is the best interpretation of the number

8

6

\newline

1

\newline

(A) The initial amount of reactant was

8

6

\newline

(B) The amount of reactant decreases by

8

6

moles per second.

\newline

(C) No reactant remains

8

6

seconds after the start of the chemical reaction.

\newline

(D) The amount of reactant begins to decrease

8

6

seconds after the start of the chemical reaction.

Gaia is driving her truck from Paris to Lyon. The function

D

gives the total distance Gaia has driven (in kilometers) after

t

\newline

What is the best interpretation for the following statement?

\newline

The slope of the line tangent to the graph of

D

t=3

90

\newline

1

\newline

(A) During the first

3

hours, Gaia drove

90

kilometers per hour.

\newline

3

, Gaia drove at a rate of

90

\newline

3

hours after leaving, Gaia was

90

kilometers away from Paris.

\newline

3

hours after leaving, Gaia drove at a rate of

90

kilometers per hour.

A weight that is attached to the end of a spring is pulled and then released. The function

H

gives its height, in centimeters, after

t

\newline

What is the best interpretation for the following statement?

\newline

H^{\prime}(0)=3

\newline

1

\newline

(A) When the weight is released, its height is

3

\newline

(B) When the weight is released, its mass is increasing at a rate of

3

grams per second.

\newline

(C) When the weight is released, its height is increasing at a rate of

3

\newline

(D) When the weight is released, its height is increasing at a rate of

3

centimeters per second.

h(t) = 2^t

change over the interval from

t = 7

t = 8

\newline

\newline

h(t)

decreases by a factor of

2

\newline

h(t)

2

\newline

h(t)

increases by a factor of

2

\newline

h(t)

t = 7

0

18=12 C+13 S

\newline

A batch of hard candy needs

C

gallons of corn syrup and

S

gallons of sucrose to create

18

pounds of sweetener as shown in the equation. What is the increase in the amount of corn syrup in gallons if the amount of sucrose decreases by

1

C=105+\frac{7}{24} m

\newline

A nutritionist uses the given equation as a measure of the total Calories,

C

, in a half bagel covered in

m

grams of cream cheese. What is the best interpretation of

24

as shown in the equation?

\newline

1

\newline

(A) A half bagel with no cream cheese has

24

\newline

(B) A half bagel and

24

grams of cream cheese have the same number of Calories.

\newline

(C) The number of Calories increases by

24

for every additional

7

grams of cream cheese.

\newline

(D) The number of grams of cream cheese increases by

24

for every additional

7

P=3.20(g-1,250)

\newline

Elena's Dry Cleaning cleans

g

garments every month. The equation gives Elena's monthly profit,

P

, in dollars, for

g

garments cleaned. What is the meaning of the

1

250

in the equation?

\newline

1

\newline

(A) Elena will make

\$ 1,250

profit if she cleans

3

2

garments per day.

\newline

(B) After cleaning

1

250

garments, Elena will have made

\$ 4,000

\newline

(C) Elena must clean

1

250

garments per month to break even with

\$ 0

\newline

(D) Elena must make

\$ 1,250

each month cleaning garments to break even with

\$ 0

S=0.014 t+0.19

\newline

Mechanical engineers determine that the maximum stress,

S

, in megapascals on a bridge with an additional mass of

t

metric tonnes uniformly spread across the bridge is given by the equation. What is the best interpretation of the expression

0.014 t

as shown in the equation?

\newline

1

\newline

(A) The portion of maximum stress due to the additional mass alone is

0.014 t

\newline

(B) The portion of maximum stress due to the mass of the bridge alone is

0.014 t

\newline

(C) The change in addtional mass associated with a

1

megapascal increase in the stress is

0.014 t

\newline

(D) The change in maximum stress associated with a

1

metric tonne increase in the mass is

0.014 t

n=38(25-d)

\newline

The number of meat patties,

n

, remaining in a restaurant freezer

d

days after restocking the freezer is approximated by the equation. What is the best interpretation of

38

as shown in the equation?

\newline

1

\newline

(A) The restaurant had

38

meat patties immediately after restocking.

\newline

(B) The restaurant has no more meat patties

38

days after restocking.

\newline

(C) The number of meat patties increases by

38

each day after restocking.

\newline

(D) The number of meat patties decreases by

38

each day after restocking.

Melissa and Rachel like to make funny hats. Melissa has made

z

zebra-printed hats, and Rachel has made

r

striped hats. Together they have made a total of

t

hats. Write an equation to describe this situation.

1790

1860

, the United States population, in thousands, is modeled by the equation

P=4000(1.031)^{t}

t

is the number of years since

1790

\newline

a. About how many people were living in the U.S. in

1790

? What about in

1860

? Show your reasoning.

Eric owns and operates the Hot Ham food truck. The expression

3.25 b+2 h

gives the cost, in dollars, of

b

h

hot dogs. What is the cost, in dollars, of

4

6

\newline

\square

Calculate the simple interest for

\mathbf{3}

years when the principal amount is

200

and interest rate is

10 \%

1

6x + 5y = 28

10x + 14y = 58

represent the cost for lunch and dinner for a family eating out on vacation. If

x

is the number of adults and

y

is the number of children, how many adults are in the family?

Q=\sqrt{\frac{2 K D}{F}}

\newline

For companies that monitor the inventory of a product, the equation gives

Q

, the quantity to order of the product as a function of

K

, the ordering cost,

D

, the annual demand for the product and

F

, the average holding cost of the product. Which of the following equations correctly gives the annual demand for the product in terms of the quantity to order, the ordering cost, the annual demand, and the average holding cost for the product?

B=11.5(132-n)

\newline

The monthly electricity bill,

B

, in dollars, for a museum using

n

solar panels is given by the equation. What is the best interpretation of

132

in the equation?

\newline

1

\newline

(A) The museum is currently using

132

\newline

(B) The monthly electric bill is

132

dollars without solar panels.

\newline

(C) The electric bill is

0

dollars when the museum uses

132

\newline

(D) The electric bill decreases by

132

dollars for each additional solar panel.