Resources

Resources

Solve quadratic equations: word problems

The area of a rectangle is

x^{2}+6 x-16

x-2

. What is a simplified expression for its length?

A robot is on the surface of Mars. The angle of depression from a camera in the robot to a rock on the surface of Mars is

13.11^{\circ}

. The camera is

181.0 \mathrm{~cm}

above the surface. How far is the camera from the rock?

\newline

\square

\mathrm{cm}

\newline

(Round to the neare tenth as needed.)

The platform height for Olympic divers is

10 \mathrm{~m}

60 \mathrm{~kg}

diver steps off the platform to begin his dive.

\newline

How much gravitational potential energy does the diver have?

Alain throws a stone off a bridge into a river below.

\newline

The stone's height (in meters above the water),

x

seconds after Alain threw it, is modeled by:

\newline

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

\newline

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

Cut Safe company manufactures scissors. The company uses the function

P

(s)=-25 s^{2}+250 s-400

to model its profit, where

P(x)

is the profit in thousands of dollars, and

x

is the number of scissors sold in tens of thousands. What is the least number of scissors the company needs to sell to break even?

h(t)=-4.9t^2+9.8t+39.2

Kaia throws a stone vertically upward from a bridge. The height, in meters, of the stone above the water seconds after the throw can be modeled by the quadratic function given. How many seconds after the throw does the stone hit the water?

A rectangle has a length of

12

meters and a width of

400

centimeters. What is the perimeter, in

\mathrm{cm}

, of the rectangle?

\newline

824

\newline

1

600

\newline

2

000

\newline

3

200

The speed of a hockey puck

x

seconds after a hockey stick strikes it is given by the function

f(x)

. The speed is measured in feet per second.

\newline

f(0.5) = 150

\newline

\newline

0.5

seconds after being struck, the puck is moving

150

feet per second.

\newline

0.5

seconds after being struck, the puck has traveled

150

Every summer, Hazelton has a music festival in the town square. The profit, in dollars, made from selling

x

tickets is given by the function

f(x)

\newline

f(200) = 1,000

\newline

\newline

(A)Hazelton can make

\$1,000

in profit by selling

200

\newline

(B)Hazelton can make

\$200

in profit by selling

1,000

The speed of a hockey puck

x

seconds after a hockey stick strikes it is given by the function

f(x)

. The speed is measured in feet per second.

\newline

f(0.5) = 150

\newline

\newline

0.5

seconds after being struck, the puck has traveled

150

\newline

0.5

seconds after being struck, the puck is moving

150

feet per second.

Every summer, Hazelton has a music festival in the town square. The profit, in dollars, made from selling

x

tickets is given by the function

f(x)

f(200) = 1,000

\newline

\newline

(A) Hazelton can make

\$200

in profit by selling

1,000

\newline

(B) Hazelton can make

\$1,000

in profit by selling

200

The perimeter of a rectangular rug is

2

4

meters. The length of the rug is

0

8

meters. What is the width in meters of the rug?

\newline

0.4 \mathrm{~m}

\newline

0.8 \mathrm{~m}

\newline

3 \mathrm{~m}

\newline

1.6 \mathrm{~m}

Members of the swim team want to wash their hair. The bathroom has less than

5600

liters of water and at most

2.5

liters of shampoo.

\newline

70L + 60S < 5600

represents the number of long-haired members

L

and short-haired members

S

who can wash their hair with less than

5600

liters of water.

\newline

0.02L + 0.01S \leq 2.5

represents the number of long-haired members and short-haired members who can wash their hair with at most

2.5

liters of shampoo.

\newline

Does the bathroom have enough water and shampoo for

8

long-haired members and

7

short-haired members?

A pool and deck are in the shape of a rectangle that is

30

\newline

- The deck is a square with a perimeter of

32

\newline

- The deep end of the pool is a square with a perimeter of

32

\newline

- The rest of the pool has shallow water.

\newline

What is the area of the part of the pool that has shallow water?

\newline

352

\newline

480

\newline

176

\newline

188

A memory chip is being designed to hold a number of transistors and heat sinks. The transistors hold memory while the heat sinks cool the chip. There must be at least one heat sink for every

2,000

transistors to prevent overheating. Also, each transistor has an area of

2.0 \times 10^{-10}\text{mm}^{2}

(square millimeters), each heat sink has an area of

3.6 \times 10^{-6}\text{mm}^{2}

, and the total area of transistors and heat sinks must be at most

2\text{mm}^{2}

. What is the approximate maximum number of transistors that the chip can hold according to this design?

\newline

1

\newline

2.78 \times 10^{2}

\newline

5.56 \times 10^{5}

\newline

1.0 \times 10^{9}

\newline

1.0 \times 10^{10}

The population of a town, in thousands, is modeled by the equation

p(t)=50 \cdot (1.06)^{t}

t

is time measured in years.

\newline

About how many years will it take for the population to double? Round to the nearest year. Explain how you know

Mustafa is staring at a pizza spinning on a wheel in a glass case.

\newline

The pizza is cut into four even slices. The distance

P(t)

\mathrm{cm}

) between the center of the tastiest looking slice and the door of the glass case as a function of time

t

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

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

\newline

t=0

, the center of the tastiest looking slice is farthest from the door, at a distance of

30 \mathrm{~cm}

2 \pi

seconds, it is closest to the door, at a distance of

10 \mathrm{~cm}

\newline

P(t)

\newline

t

should be in radians.

\newline

P(t)=

The speed of sound in air is about

332

meters per second

\left(\frac{\mathrm{m}}{\mathrm{s}}\right)

0

degrees Celsius

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

. If the speed increases by

0.6 \frac{\mathrm{m}}{\mathrm{s}}

for every increase in temperature of

1^{\circ} \mathrm{C}

, which inequality best represents the temperatures,

T

, in degrees Celsius, for which the speed of sound in air exceeds

350 \frac{\mathrm{m}}{\mathrm{s}}

3-4-5

right triangle is inscribed in circle

A

5-12-13

right triangle is inscribed in circle

B

. What is the ratio of the area of circle

A

to the area of circle

B

Pluto's distance from the sun varies in a periodic way that can be modeled approximately by a trigonometric function.

\newline

Pluto's maximum distance from the sun (aphelion) is

7.4

billion

km

. Its minimum distance from the sun (perihelion) is

4.4

billion

km

. Pluto last reached its perihelion in the year

1989

, and will next reach perihelion in

2237

\newline

Find the formula of the trigonometric function that models Pluto's distance

D

from the sun (in billion km)

t

2000

. Define the function using radians.

\newline

D(t)=5.9+1.5 \cos\left(\frac{2\pi}{248}(t+11)\right)

\newline

How far will Pluto be from the sun in

2000

? Round your answer, if necessary, to two decimal places.

\newline

billion

km

100

meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width

x

(in meters) is modeled by:

A(x) = -(x - 25)^2 + 625

What is the maximum area possible?

Lillian has had enough of Tiffany's antics and runs over to Tiffany's table. With an initial speed of

2\,\text{m/s}

, Lillian accelerates at a rate of

18\,\text{m/s}^2

1.5

seconds, what is her final speed?

\newline

29\,\text{m/s}

\newline

35\,\text{m/s}

\newline

19\,\text{m/s}

\newline

23\,\text{m/s}

A ball is thrown up into air from the ground and the ball's velocity is modeled by the equation

v(t)=-\frac{1}{4}x^{2}-4\cos(\pi x)+8

\newline

t

value does the ball reach its maximum?

Gabriel is designing a rectangular planter box for their garden. It needs to cover an area of

15 \frac{1}{2} \mathrm{~m}^{2}

. Gabriel wants it to be

7 \frac{3}{4} \mathrm{~m}

\newline

How wide does the planter box need to be?

\newline

\_\_\_\_\_

The equation of line

f

y+3=\frac{1}{6}(x-2)

. Parallel to line

f

g

, which passes through the point

(-6,1)

. What is the equation of line

g

\newline

Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

Tacoma Teco Cart inc. le looking to purchace a new taco cart to expand operatione, at a cost of

\$109,000

. Tacome desires an Internal rate of return on this new cart of

11\%

and estimates it will have a useful life of

5

yeare. Tacome usee the Present Value of an Annulity of

\$1

table to determine an annulity factor of

3.6959

. Given these parameters, how much yearly cash flow would this taco cart need to produce?

\newline

\$29,492.14

\newline

\$40,285.31

\newline

\$21,800.00

\newline

\$35,639.20

Rock Hill Relaxation Company sells vitamin supplements. They have collected the following budgot information about the

10

000

bottles of supplemente they expect to sell; Total Direct Material Cost

\$ 27,000

, Total Direct Labor Cost

\$ 15,000

, Total Manufacturing Overhead Cost

\$ 12,000

, Total Selling and Administrative Expenees

\$ 3,000

. Rock Hill's desired profit is a

50 \%

return on investment of their

\$ 90,000

of total assets.

\newline

At what amount should Rock Hill price its supplements, II they are using the Product Cost Method of prieing?

\newline

\$ 5.30

\newline

\$ 10.70

\newline

\$ 15.75

\newline

\$ 13.65

A prism has a height of

5

inches and a volume of

80

cubic inches. Select all figures that could be the base for this prism.

\newline

2

-8

-inch rectangle

\newline

B. A circle with radius

2

\newline

C. A heart-shaped base with an area of

16

\newline

D. A square with side length

4

\newline

E. A right triangle with legs

4

The kinetic energy, measured in kilogram meters squared per second squared

\left(\frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}\right)

, of the International Space Station is approximately:

\newline

13,340,250,000,000 \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}

\newline

1

J

1 \frac{\text{kg} \cdot \text{m}^2}{\text{s}^2}

1

TJ

10^{12}J

, what is the approximate kinetic energy of the Space Station in terajoules to the nearest hundredth?

\newline

1

\newline

0.1334\,TJ

\newline

1.334\,TJ

\newline

13.34\,TJ

\newline

13.34 \times 10^{12}\,TJ

Andres kicks a football. Its height in feet is given by

h(t)=-16 t^{2}+64 t

t

represents the time in seconds after kick. After how many seconds does the football hit the ground?

\newline

Answer: seconds

Avani launches a toy rocket from a platform. The height of the rocket in feet is given by

h(t)=-16 t^{2}+16 t+32

t

represents the time in seconds after launch. What is the rocket's greatest height?

\newline

Liam launches a toy rocket from a platform. The height of the rocket in feet is given by

h(t)=-16 t^{2}+40 t+96

t

represents the time in seconds after launch. How many seconds have gone by when the rocket is at its highest point?

\newline

Answer: seconds

Rajani kicks a football. Its height in feet is given by

h=-16 t^{2}+24 t

t

represents the time in seconds after kick. How many seconds have gone by when the football is at its highest point?

\newline

Answer: seconds

A tennis ball is dropped from a certain height. Its height in feet is given by

h(t)=-16 t^{2}+20

t

represents the time in seconds after launch. What is the ball's initial height?

\newline

A tennis ball is dropped from a certain height. Its height in feet is given by

h=-16 t^{2}+24

t

represents the time in seconds after launch. What is the ball's initial height?

\newline

A rocket is launched in the air. Its height in feet is given by

h=-16 t^{2}+96 t

t

represents the time in seconds after launch. How many seconds have gone by when the rocket is at its highest point?

\newline

Answer: seconds

Jeriel launches a toy rocket from a platform. The height of the rocket in feet is given by

h(t)=-16 t^{2}+120 t+136

t

represents the time in seconds after launch. How many seconds have gone by when the rocket is at its highest point?

\newline

Answer: seconds

A mug of warm apple cider is gradually cooling. Its temperature in degrees Celsius

({}^\circ C)

can be modeled with the expression

23+24\cdot 2^{-0.014 t}

, where the variable

t

represents the time in minutes. Approximately how many minutes will it take for the temperature to reach

29{}^\circ C

? (Round to the nearest minute.)

S=16 t^{2}

is used to approximate the distance

S

, in feet, that an object falls freely from rest in

t

seconds. The height of a building is

1390

feet. How long would it take for an object to fall from the top?

The perimeter of a rectangle is

46

meters. Find the length and width if the length is an integer and the width is

2

times the next consecutive integer.

\newline

\text { Length }=\square \text { meters } \quad \text { Width }=\square \text { meters }

The perimeter of a rectangle is

142

meters. Find the length and width if the length is an integer and the width is

2

times the next consecutive integer.

\newline

\text { Length }=\square \text { meters } \quad \text { Width }=\square \text { meters }

An object is launched from a platform. Its height (in meters),

x

seconds after the launch, is modeled by:

\newline

h(x)=-5x^{2}+20x+60

\newline

What is the height of the object at the time of launch?

The given equation models the revenue in thousands of dollars,

R

t

2012

. Of the following, which equation models the revenue of the company

q

2012

\newline

1

\newline

R=380(1.02)^{4q}

\newline

R=380(1.36)^{q}

\newline

R=380(1.08)^{\frac{q}{4}}

\newline

R=380(1.08)^{4q}

The formula gives the temperature change

r

, in degrees Celsius per minute, of tortellini while it is being cooked, given the initial temperatures of the tortellini,

T_{t}

, and the water,

T_{w}

, in degrees Celsius, and a heat conductivity of the tortellini of

k

joules per square centimeter per second. Which equation correctly gives the heat conductivity of the tortellini in terms of the temperature change per minute and the initial temperatures of the water and tortellini?

\newline

1

\newline

k=\frac{200(r+T_{w})}{-181T_{t}}

\newline

k=\frac{200r}{-181(T_{t}-T_{w})}

\newline

k=\frac{200r+T_{W}}{-181T_{t}}

An ice sculpture in the form of a sphere melts in such a way that it maintains its spherical shape. The volume of the sphere is decreasing at a constant rate of

2\pi

cubic meters per hour. At what rate, in square meters per hour, is the surface area of the sphere decreasing at the moment when the radius is

5

meters? (Note: For a sphere of radius

r

, the surface area is

4\pi r^{2}

and the volume is

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

\newline

\frac{4\pi}{5}

\newline

40\pi

\newline

80\pi^{2}

\newline

100\pi

The atmospheric pressure of the air changes with height above sea level. The pressure of the air at a given height above sea level can be measured by the differentiable function

f(h)

, in psi, where

h

is measured in kilometers. What are the units of

\int_{0}^{5} f^{\prime}(h) d h ?

\newline

\newline

\newline

psi / kilometer

\newline

kilometers / psi

\newline

psi / kilometer

{ }^{2}

\newline

/ \mathrm{psi}^{2}

In a particular city, the amount of snow on the ground measured by the differentiable function

f

f(t)

is measured in centimeters and

t

is measured in minutes after the beginning of a snow storm. What are the units of

\int_{2}^{10} f^{\prime}(t) d t

\newline

\newline

\newline

centimeters / minute

\newline

minutes / centimeter

\newline

centimeters / minute

{ }^{2}

\newline

minutes / centimeter

{ }^{2}

The rate at which people are entering a line to buy a new piece of technology is measured by the differentiable function

f

f(t)

is measured in people per minute and

t

is measured in minutes after the store opened. What are the units of

\frac{1}{5} \int_{3}^{8} f^{\prime}(t) d t ?

\newline

\newline

\newline

minutes / person

\newline

people / minute

\newline

minutes / person

{ }^{2}

\newline

people / minute

{ }^{2}

The differentiable function

f

measures the rate at which the temperature of a turkey in an oven is changing, in degrees Fahrenheit per hour, where

t

is measured in hours. What are the units of

f^{\prime}(t)

\newline

degrees Fahrenheit

\newline

degrees Fahrenheit / hour

\newline

degrees Fahrenheit / hour

^{2}

\newline

\newline

hours / degree Fahrenheit

\newline

hours / degree Fahrenheit

{ }^{2}

The position of a bicycle riding down a straight road is measured by the differentiable function

f

f(t)

is measured in meters and

t

is measured in seconds. What are the units of

\frac{1}{6} \int_{3}^{9} f(t) d t

\newline

\newline

\newline

seconds / meter

\newline

meters / second

\newline

/

^{2}

\newline

meters / second

{ }^{2}