Resources

Resources

Interpret parts of quadratic expressions: word problems

To get over a fence, a cat jumps with an initial upward velocity of

20

feet per second and then lands back on the ground. Therefore, the cat's height above the ground in feet,

t

seconds after it jumps, can be modeled by the expression

-16t^2 + 20t

. This expression can be written in factored form as

-16t(t - 1.25)

\newline

What does the number

1.25

represent in the expression?

\newline

(A)the height in feet of the cat at the start of its jump

\newline

(B)the height in feet of the cat when it reaches its highest point

\newline

(C)the time in seconds from when the cat jumps until it lands on the ground

\newline

(D)the time in seconds from when the cat jumps until it reaches its highest point

On a windy autumn day, an acorn falls from a branch of a giant oak tree. The acorn's height above the ground in feet can be modeled by the expression

80 - 16t^2

t

is the time in seconds after the acorn begins to fall.

\newline

What does the quantity

16t^2

represent in the expression?

\newline

\newline

(A)the height in feet of the acorn after

t

\newline

(B)the distance in feet the acorn has fallen after

t

\newline

(C)the time in seconds it takes for the acorn to fall

t

\newline

(D)the time in seconds it takes for the acorn to reach a height of

t

\newline

\pi

is defined as the ratio of the circumference (say

c

) of a circle to its diameter (say

d

\pi=\frac{c}{d}

. This seems to contradict the fact that

\pi

is irrational. How will you resolve this contradiction?

Amir stands on a balcony and throws a ball to his dog, who is at ground level.

\newline

The ball's height (in meters above the ground),

x

seconds after Amir threw it, is modeled by:

\newline

h(x)=-(x-2)^{2}+16

\newline

What is the height of the ball at the time it is thrown?

\newline

\square

Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground),

t

seconds after Amir threw it, is modeled by How many seconds after being thrown will the ball reach its maximum height?

t

Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground),

x

seconds after Amir threw it, is modeled by:

h(x) = -(x - 2)^2 + 16

. How many seconds after being thrown will the ball hit the ground?

Alain throws a stone off a bridge into a river below. The stone's height (in meters above the water),

x

seconds after Alain threw it, is modeled by:

h(x)= -5x^2+10x+15

. How many seconds after being thrown will the stone hit the water?

\text{seconds}

The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:

m(x)=-x^2+14x

What is the maximum possible number of mosquitoes?

\_\_\_\_\_

million mosquitoes

Julian is using a biking app that compares his position to a simulated biker traveling Julian's target speed. When Julian is behind the simulated biker, he has a negative position.

\newline

Julian sets the simulated biker to a speed of

20 \frac{\mathrm{km}}{\mathrm{h}}

. After he rides his bike for

15

minutes, Julian's app reports a position of

-2 \frac{1}{4} \mathrm{~km}

\newline

What has Julian's average speed been so far?

\newline

\frac{\mathrm{km}}{\mathrm{h}}

2

granola bars and drinks

1

glass of orange juice for breakfast every day. The price of a granola bar is

B

dollars and the price of a glass of orange juice is

J

\newline

7(2 B+J)

describes how much Donovan spends on breakfast in one week.

\newline

2 B

represent in the expression?

\newline

1

\newline

(A) Amount Donovan spends on breakfast each day

\newline

(B) Amount Donovan spends on granola bars each day

\newline

(C) Amount Donovan spends on juice each day

\newline

(D) Number of days Donovan buys breakfast.

12 g

gives the number of kilometers a car can travel using

g

liters of gasoline.

\newline

How far can this car travel on

5 \frac{1}{2}

liters of gasoline?

\newline

\mathrm{km}

A caterpillar eats

1400 \%

of its birth mass in one day. The caterpillar's birth mass is

m

\newline

Which of the following expressions could represent the amount, in grams, the caterpillar eats in one day?

\newline

2

\newline

14 m

\newline

1.4 m

\newline

1400 m

\newline

\frac{1400}{100} m

\newline

140 m

After learning how to fly, Wendy reduced her daily commute time by

75 \%

. Previously, her commute took

m

\newline

Which of the following expressions could represent Wendy's commute time, in minutes, after she learned how to fly?

\newline

2

\newline

0.25 m

\newline

0.75 m

\newline

1.75 \mathrm{~m}

\newline

\frac{1}{4} m

\newline

\frac{3}{4} m

Vlad is playing on a swing set.

\newline

His horizontal distance

D(t)

(in m) from the center (where being behind the center means a negative distance) as a function of time

t

(in seconds) can be modeled by a sinusoidal expression of the form

a \cdot \cos (b \cdot t)+d

\newline

t=0

, when he pushes off, he is

1 \mathrm{~m}

behind the center, which is as far back as he goes. The swing reaches the center

\frac{\pi}{6}

\newline

D(t)

\newline

t

should be in radians.

\newline

D(t)=\square

Scarlett sells designer ponchos. The revenue,

R

, she makes from selling a batch of

n

ponchos is given by the function

R(n)=20 n

, and the expenses,

E

, that she incurs from producing a batch of

n

ponchos is given by the function

\newline

E(n)=10 \cdot \sqrt{n} \text {. }

\newline

P

be the total profit Scarlett makes from producing and selling a batch of

n

\newline

Write a formula for

P(n)

R(n)

E(n)

\newline

P(n)=

\newline

Write a formula for

P(n)

n

\newline

P(n)=\square

If all goes according to plan, the function

A(t)=200(1.1)^{t}

will model the amount of potatoes,

A

, in bushels, produced by Burian's farm

t

years from now, and the function

R(t)=5000(1.111)^{t}

will model the revenue,

R

, in dollars, earned from selling these potatoes.

\newline

P

be the proposed price of a single bushel of potatoes

t

years from now.

\newline

Write a formula for

P(t)

A(t)

R(t)

\newline

P(t)=

\newline

Write a formula for

P(t)

t

\newline

P(t)=

Walking on his own, the distance,

D

, in feet, that Roberto can cover in

m

minutes is given by the function

D(m)=264 m

. When he walks on the moving sidewalk at the airport, the distance,

A

, in feet, that he can cover in

m

minutes is given by the function

\newline

A(m)=440 m \text {. }

\newline

B

be the distance, in feet, that Roberto would travel on the moving sidewalk in

m

minutes if he were standing still.

\newline

Write a formula for

B(m)

D(m)

A(m)

\newline

B(m)=

\newline

Write a formula for

B(m)

m

\newline

B(m)=\square

Julie is measuring two cones.

\newline

Given the base radius

r

h

of the first cone, Julie uses the formula

\newline

V=\frac{\pi r^{2} h}{3}

\newline

to compute its volume

V

6

\newline

The second cone has the same height, but has

2

times the radius. What is its volume?

\newline

1

\newline

12

\newline

4 \pi

\newline

24

\newline

8 \pi

Avery is measuring two cylinders.

\newline

Given the volume

V

and the base radius

r

of the first cylinder, Avery uses the formula

\newline

h=\frac{V}{\pi r^{2}}

\newline

to compute its height

h

24

\newline

The second cylinder has the same volume, but has

2

times the radius. What is its height?

\newline

1

\newline

6

\newline

12

\newline

48

\newline

96

Bjorn is making aush, and he does not have enough of all the ingredients. He decides to make

\frac{5}{6}

of a whole recipe, which means he uses

1 \frac{1}{4} \mathrm{~L}

\newline

How much water is in a whole recipe?

\newline

2020

6

000

randomly selected youths ages

13

17

9

5

percent identified as LGBT. Based on the results of the survey, which of the following are reasonable statements?

\newline

9

5

percent of the US population identify as LGBT.

\newline

9.5 \%

13

17

in the US identify as LGBT.

\newline

III. If another

6

000

randomly selected youths ages

13

17

in the US were surveyed, about

9

5

percent of them would identify as LGBT.

\newline

1

\newline

\newline

\newline

(C) I and II only

\newline

(D) II and III only

z=44 i-32

\newline

What are the real and imaginary parts of

z

\newline

1

\newline

\newline

\begin{array}{l} \operatorname{Re}(z)=44 \text { and } \\ \operatorname{Im}(z)=-32 \end{array}

\newline

\newline

\begin{array}{l} \operatorname{Re}(z)=-32 \text { and } \\ \operatorname{Im}(z)=44 i \end{array}

\newline

\newline

\begin{array}{l} \operatorname{Re}(z)=44 i \text { and } \\ \operatorname{Im}(z)=-32 \end{array}

\newline

\newline

\begin{array}{l} \operatorname{Re}(z)=-32 \text { and } \\ \operatorname{Im}(z)=44 \end{array}

M(t)

models the distance (in millions of

\mathrm{km}

) from Mars to the Sun

t

days after it's at its furthest point.

\newline

M(t)=21 \cos \left(\frac{2 \pi}{687} t\right)+228

\newline

What does the solution set for

240=21 \cos \left(\frac{2 \pi}{687} t\right)+228

\newline

1

\newline

(A) The farthest distance, in millions of kilometers, that Mars reaches from the sun

\newline

(B) The least number of days after when Mars is at its furthest point when it is

240

million kilometers from the sun

\newline

(C) The distance, in millions of kilometers, that Mars is from the sun after

240

A study suggests that

75 \%

of household solid waste can be recycled. However, the average person only recycles

30 \%

of his or her solid waste. If the average person produces

4

5

pounds of solid waste a day, approximately how many more pounds of solid waste could be recycled compared to the amount they currently recycle?

\newline

1

\newline

1

35

\newline

2

03

\newline

3

38

\newline

\mathbf{1 1 . 2 5}

Beverly surveyed a random sample of employees at Bloblin, Inc. on whether they prefer to work remotely at least

2

days a week. The results of the survey are most representative of which of the following populations?

\newline

1

\newline

(A) Employees at companies with a similar number of employees as Bloblin, Inc.

\newline

(B) Employees at Bloblin, Inc.

\newline

(C) Employees at Bloblin, Inc. who work remotely

\newline

(D) Employees at Bloblin, Inc. who do not work remotely

The mass of Stewart's favorite frying pan is

0

52

kilograms. What is the mass of the frying pan in grams?

\newline

1

=1,000

\newline

1

\newline

5

2

\newline

52

\newline

520

\newline

5

200

Lea's car travels an average of

30

miles per gallon of gas. If she spent

\$ 20.70

172

5

mile trip, what was the approximate cost of gas in dollars per gallon?

\newline

1

\newline

\$ 1.45

\newline

\$ 3.40

\newline

\$ 3.60

\newline

\$ 5.75

The formula for the remaining volume of fuel in a car's tank is

I-E \cdot D

I

is the initial volume of fuel,

E

is the fuel efficiency, and

D

is the distance traveled.

\newline

Carson drove a distance of

120

kilometers. He initially had

30

liters of fuel, and his car's fuel efficiency is

100

cubic centimeters per kilometer.

\newline

What calculation will give us the estimated volume of fuel that remains in Carson's tank by the end of the drive, in liters?

\newline

1

\newline

30-\frac{100}{1000} \cdot 120

\newline

30 \cdot 1000-100 \cdot 120

\newline

\frac{30}{1000}-100 \cdot 120

\newline

30-100 \cdot 1000 \cdot 120

15

kilometers in one hour.

\newline

How many kilometers can Amir run per minute?

\newline

60

1

\newline

1

\newline

0

25

\newline

4

\newline

900

Emanuel surveyed a random sample of

50

subscribers to Auto Wheel magazine about the number of cars that they own. Of the subscribers surveyed,

15

2

vehicles. There are

340

subscribers to Auto Wheel magazine. Based on the data, what is the most reasonable estimate for the number of Auto Wheel magazine subscribers who own fewer than

2

\newline

1

\newline

15

\newline

84

\newline

102

\newline

238

h=3.2 \ell+82.77

\newline

An anthropologist uses the formula to estimate the height,

h

, in centimeters, of a male with an ulna bone that is

\ell

centimeters long. Which of the following equations correctly gives the length of the ulna bone in terms of the height?

\newline

1

\newline

\ell=\frac{h}{3.2}-82.77

\newline

\ell=\frac{h-82.77}{3.2}

\newline

\ell=3.2 h-82.77

\newline

\ell=h-\frac{82.77}{3.2}

C=1.2272 M+3.0556 F

\newline

An English teacher uses the equation above to give the composite score,

C

, on the final exam, given

M

correct multiple choice answers and

F

points on the free response questions. Which of the following equations correctly gives the number of correct multiple choice answers in terms of the composite score and the number of points on the free response questions?

\newline

1

\newline

M=\frac{C-3.0556 F}{1.2272}

\newline

M=\frac{C}{1.2272}-3.05561

\newline

M=\frac{3.0556 F-C}{1.2272}

\newline

M=\frac{3.0556 F}{1.2272}-C

V=\frac{4}{3} \pi r^{3}

\newline

The formula gives the volume

V

of a sphere with radius

r

. What is the volume, in cubic feet, of a sphere with a radius of

2

\newline

1

\newline

4 \pi

\newline

8 \pi

\newline

\frac{8}{3} \pi

\newline

\frac{32}{3} \pi

\frac{1}{2} b h

gives the area of a triangle, where

b

is the base of the triangle and

h

is the height of the triangle. What is the area, in square centimeters, of a triangle with a base of

7

centimeters and a height of

4

The price, in dollars, of a visit to the dentist is calculated according to the formula

50+100 n

n

is the number of cavities the dentist finds. On Vanessa's last visit, the dentist found

2

cavities. What was the cost, in dollars, of Vanessa's visit?

Cell phone provider A charges

\$ 60

1

gigabyte (GB) of data plus

\$ 0.05

for each megabyte (MB) of overage data. Cell phone provider B charges

\$ 45

1 \mathrm{~GB}

\$ 0.10

for each MB of overage data. Assuming there will be data overage, how many MB of overage data would make the cost of both data plans the same?

40^{\circ} \mathrm{F}

25

(\mathrm{mph})

wind creates a wind chill of

29^{\circ} \mathrm{F}

. To approximate the wind chill at various temperatures caused by a

25 \mathrm{mph}

wind, Elon uses the following rule: take

24

\frac{4}{3}

of the air temperature

\left({ }^{\circ} \mathrm{F}\right)

25 \mathrm{mph}

wind creates a wind chill of

-5^{\circ} \mathrm{F}

, which of the following best approximates the corresponding air temperature?

\newline

1

\newline

-30.7^{\circ} \mathrm{F}

\newline

14.3^{\circ} \mathrm{F}

\newline

(\mathrm{mph})

0

\newline

(\mathrm{mph})

1

John manufactures household furniture. His start-up costs, including tools, plans, and advertising, total

\$ 5000

. Labor and materials for each piece of furniture total

\$ 350

, and his production costs are the sum of his start-up costs plus the cost of labor and materials. If John makes

f

pieces of furniture, what will be his production costs,

c

\newline

1

\newline

f=350 c+5000

\newline

c=350 f+5000

\newline

f=5000 c+350

\newline

c=5000 f+350

The number of problems assigned as homework by a savvy math professor is a function of the length of the class on a particular day, which is either

45

90

135

minutes. He first multiplies the number of minutes by

4

and then divides that number by

9

. Finally, he subtracts

12

from that amount. Which of the following functions best represents the number of math problems the professor assigns depending on the length of the class,

t

\newline

1

\newline

f(t)=\frac{4}{9} t-12

\newline

f(t)=\frac{4}{9}(t-12)

\newline

f(t)=12-\frac{4}{9} t

\newline

f(t)=12+\frac{4}{9} t

A roller coaster is currently traveling at a speed of

49

miles per hour (mph). The coaster's speed will increase at a constant rate of

17 \mathrm{mph}

2

seconds until the coaster reaches its top speed

5

seconds from now. If

t \leq 5

, which function best represents the roller coaster's speed, in miles per hour,

t

seconds from now?

\newline

1

\newline

f(t)=49+8.5 t

\newline

f(t)=49+17 t

\newline

f(t)=49 t+9.8 t

\newline

f(t)=49 t+17 t

A cleaner is sold in low and high concentrations so that customers can combine amounts from each in order to obtain a desired quantity and concentration. The low concentration is

3 \%

pure cleaner and the high concentration is

18 \%

pure cleaner. How many liters of the low and high concentrations must be combined to obtain

10

8 \%

\newline

1

\newline

6

liters of low and

4

\newline

6 \frac{1}{3}

liters of low and

3 \frac{2}{3}

\newline

6 \frac{2}{3}

liters of low and

3 \frac{1}{3}

\newline

7

liters of low and

3

Ruby can assemble

2

gift baskets by herself in

7

minutes. Emma can assemble

4

gift baskets by herself in

15

minutes. Ruby begins assembling gift baskets at

1: 00

\mathrm{p} . \mathrm{m}.,

and Emma begins assembling gift baskets at

1: 15

\mathrm{p} . \mathrm{m}.

If they continue to work at the above rates, at what time will they finish the

54^{\text {th }}

\newline

1

\newline

2: 30

\mathrm{p} . \mathrm{m}

\newline

2: 42

\mathrm{p} . \mathrm{m}

\newline

2: 45

\mathrm{p} . \mathrm{m}

\newline

3: 00

\mathrm{p} . \mathrm{m}

From \(1970\) to \(1990\) the average per person annual consumption of whole milk declined at a constant rate from \(101\) to \(41\) quarts per year. During the same period, the average annual consumption of low fat milk increased at a constant rate from \(25\) to \(60\) quarts per year. What was the average annual per person consumption when the consumption of whole milk and low fat milk was the same?

\newline

1

\newline

(A) \(53\) quarts

\newline

(B) \(55\) quarts

\newline

(C) \(57\) quarts

\newline

(D) \(59\) quarts

A caterer is determining how many forks she will need to buy for her upcoming event. Each adult needs \(5\) forks, and each child needs \(2\) forks. If the event will host \(764\) adults and children in all, and the caterer ordered

2,992

forks, how many adults and how many children are expected to attend?

\newline

1

\newline

(A) \(207\) adults and \(557\) children

\newline

(B) \(276\) adults and \(488\) children

\newline

(C) \(488\) adults and \(276\) children

\newline

(D) \(557\) adults and \(207\) children

Uche is a cartographer. He picks a scale to fit a map of India onto a page of an atlas. The page is

12

12

0

75

inch margins on all

4

sides. India measures

3

214

kilometers from north to south and

2

933

kilometers from west to east. Uche wants the longest dimension of India to fit exactly in between the margins of the page. If

k

is the number of kilometers per inch in Uche's scale, which equation best models the situation?

\newline

1

\newline

\frac{k}{12}=3,214

\newline

12 k=2,933

\newline

\frac{k}{10.5}=2,933

\newline

10.5 k=3,214

A farmer plans to install solar collection panels to provide winter heating for the livestock. The most efficient panel angle for winter heating in the farm's region is

60^{\circ}

relative to ground level. If an individual panel is

67

inches (in) long and installed on the ground according to these instructions, what is the height, in in, of the upper edge above the ground?

\newline

1

\newline

33.5 \sqrt{3}

\mathrm{in}

\newline

33.5 \sqrt{2}

\mathrm{in}

\newline

33.5

\mathrm{in}

\newline

\frac{134 \sqrt{3}}{3}

\mathrm{in}

Wanahton is cooking a breadstick on a rectangular baking sheet measuring

9 \frac{1}{2}

\mathrm{in}

13

\mathrm{in}

. Assuming the breadstick width is negligible, what is the longest breadstick Wanahton could bake by fitting it straight along the diagonal and within the baking sheet to the nearest inch?

\newline

1

\newline

13

\mathrm{in}

\newline

16

\mathrm{in}

\newline

124

\mathrm{in}

\newline

259

\mathrm{in}

Erica walked down a trail into a valley.

\newline

V(d)

models Erica's vertical distance from the top of the valley (in kilometers) after walking

d

kilometers along the trail.

\newline

What does the statement

V(2)=T

\newline

1

\newline

(A) Erica's vertical distance from the top of the valley was the same after she had walked

2

km and after she had walked

T

\newline

(B) After she had walked

2

km along the trail, Erica's vertical distance from the top of the valley was equal to

T

\newline

(C) After she had walked

T

kilometers along the trail, Erica's vertical distance from the top of the valley was equal to

2

Tessa's class had a math exam where the grades were between

0

10

\newline

N(g)

models the number of students whose grade on the exam was

g

\newline

What does the statement

N(8) > 2 \cdot N(5)

\newline

1

\newline

8

students whose grade was higher than twice the grade of another group of

5

\newline

(B) The number of students whose grade was

8

is more than twice the number of students whose grade was

5

\newline

(C) The number of students whose grade was higher than

5

is greater than the number of students whose grade was

5

Erica walked down a trail into a valley.

\newline

V(d)

models Erica's vertical distance from the top of the valley (in kilometers) after walking

d

kilometers along the trail.

\newline

What does the statement

V(2)=T

\newline

1

\newline

(A) Erica's vertical distance from the top of the valley was the same after she had walked

2

km and after she had walked

T

\newline

(B) After she had walked

2

km along the trail, Erica's vertical distance from the top of the valley was equal to

T

\newline

(C) After she had walked

T

kilometers along the trail, Erica's vertical distance from the top of the valley was equal to

2