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Area of quadrilaterals and triangles: word problems

The radius of a circle is increasing at a rate of

3

centimeters per second.

\newline

At a certain instant, the radius is

8

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

\newline

1

\newline

192 \pi

\newline

64 \pi

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9 \pi

\newline

48 \pi

The radius of a sphere is increasing at a rate of

7

5

meters per minute.

\newline

At a certain instant, the radius is

5

\newline

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

\newline

1

\newline

100 \pi

\newline

330 \pi

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300 \pi

\newline

225 \pi

\newline

The surface area of a sphere with radius

r

4 \pi r^{2}

The radius of the base of a cylinder is decreasing at a rate of

9

millimeters per hour and the height of the cylinder is increasing at a rate of

2

millimeters per hour.

\newline

At a certain instant, the base radius is

8

millimeters and the height is

3

\newline

What is the rate of change of the surface area of the cylinder at that instant (in square millimeters per hour)?

\newline

1

\newline

155 \pi

\newline

310 \pi

\newline

-155 \pi

\newline

-310 \pi

\newline

The surface of a cylinder with base radius

r

h

2 \pi r h+2 \pi r^{2} \text {. }

The radius of the base of a cylinder is decreasing at a rate of

9

millimeters per hour and the height of the cylinder is increasing at a rate of

2

millimeters per hour.

\newline

At a certain instant, the base radius is

8

millimeters and the height is

3

\newline

What is the rate of change of the surface area of the cylinder at that instant (in square millimeters per hour)?

\newline

1

\newline

310 \pi

\newline

-310 \pi

\newline

155 \pi

\newline

-155 \pi

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The surface of a cylinder with base radius

r

h

2 \pi r h+2 \pi r^{2} \text {. }

A pendulum is swinging back and forth. After

t

seconds, the horizontal distance from the bob to the place where it was released is given by

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H(t)=7-7 \cos \left(\frac{2 \pi(t-2)}{20}\right) .

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How often does the bob cross its midline? Give an exact answer.

\newline

\square

A piece of paper is to display

128

square inches of text. If there are to be one-inch margins on both sides and two-inch margins at the bottom and top, what are the dimensions of the smallest piece of paper (by area) that can be used?

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1

\newline

8 \prime \prime \times 16 \prime \prime

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10 \prime \prime \times 15 \prime \prime

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10 \prime \prime \times 18 \prime \prime

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10 \prime \prime \times 20 \prime \prime

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(E) None of these

A(x) = (8 + 2x)^2

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Carmen wants to add a border around a square picture. The function shows the area of the picture in square inches if it has a border

x

inches wide. What are the dimensions of the picture without the border?

Calculate the area of the rectangle when

a=4

b=2

. Next, calculate the area when the length and width are doubled. Does the area double? What is the ratio of the two areas you calculated?

\newline

1:4

\newline

1:4

\newline

1:2

\newline

1:2

s^{2}

is used to calculate the areag of a square, where

s

is the side length of the square. What does the expression

(8 x)^{2}

\newline

(A) the area of a square with a side length of

8

\newline

(B) the area of a square with a side length of

16

\newline

(C) the area of a square with a side length of

4 x

\newline

(D) the area of a square with a side length of

8 x

5

) A rectangular tank of height

40 \mathrm{~cm}

had a base area of

300 \mathrm{~cm}^{2}

. It was filled with water to a depth of

30 \mathrm{~cm}

. After some water was poured out, the depth of the water decreased to

20 \mathrm{~cm}

. How much water was poured out? Give your answer in litres. (

\ell=1000 \mathrm{~cm}^{3}

Find the area of a rhombus if its vertices are

(3,0),(4,5),(-1,4)

(-2,-1)

taken in order. [Hint : Area of a rhombus

=\frac{1}{2}

(product of its diagonals)]

A gemcutter wants to trim a gemstone that is approximately a regular octahedron with sides measured in millimeters (mm). She will remove imperfections by reducing the length of each side by

x

mm. The approximate new surface area is given in

\text{mm}^2

by the function:

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S(x)=3.4(56.25-15x+x^2)

\newline

What is the current length of each side?

The rectangular floor of a classroom is

40

feet in length and

20

feet in width. A scale drawing of the floor has a length of

20

inches. What is the area, in square inches, of the floor in the scale drawing?

\newline

\square

^{2}

The rectangular floor of a classroom is

38

feet in length and

32

feet in width. A scale drawing of the floor has a length of

19

inches. What is the area, in square inches, of the floor in the scale drawing?

\newline

\square

^{2}

The rectangular floor of a classroom is

20

feet in length and

32

feet in width. A scale drawing of the floor has a length of

5

inches. What is the area, in square inches, of the floor in the scale drawing?

\newline

\square

^{2}

\overline{Q R}

\overline{S T}

are both diameters of a circle. This circle passes through the point

(3,13.2)

\overline{Q R}

lies on the line whose equation is

y=2.5 x+4.9

\overline{S T}

lies on the line whose equation is

y=3 x+2.4

, what is the area of the circle?

Camacho is buying a monster truck. The price of the truck is

x

dollars, and he also has to pay a

13 \%

monster truck tax.

\newline

Which of the following expressions could represent how much Camacho pays in total for the truck?

\newline

2

\newline

(1+0.13) x

\newline

\frac{13}{100} x

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13 x

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1.13 x

\newline

13 x+x

Last month, Marcus borrowed

\$88

from his roommate. So far, he has paid him back

\$34

. Which equation could be used to find

x

, the amount of money he still owes his roommate?

\newline

Find the exact volume and surface area of a sphere of radius

12

\newline

V=576\pi\text{ cu in.};

SA=2304\pi\text{ sq in.}

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V=2304\pi\text{ cu in.};

SA=576\pi\text{ sq in.}

\newline

V=1728\pi\text{ cu in}

;SA=144\pi\text{ sq in.}

\newline

V=6912\pi\text{ cuin.}:

SA^{\cdot}=36\pi\text{ sq in.}

\newline

A regular pentagon box has a base perimeter

P

5b^2 - 10b - 15

units and a height of

b

units. The area of a regular pentagon is

A = \frac{1}{2} aP

a =

P =

perimeter. If the apothem of the base is

2b - 4

units, what is the surface area of the pentagon box?

h=11.2-0.125d

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The ceiling height,

h

, in feet, for a particular room in a house a distance of

d

feet from the west wall is given by the equation. In order for the ceiling height to decrease by

1

foot, how much does the distance from the west wall change in feet?

\newline

1

\newline

0.125

\newline

8

\newline

11.2

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89.6

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

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(B) After she had walked

2

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

T

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(C) After she had walked

T

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

2

\frac{1}{4}

cup of sugar. How many tablespoons does the recipe use?

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\text { ( } 1 \text { cup }=16 \text { tablespoons) }

\newline

1

\newline

4

\newline

8

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32

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64

The following information is available for them: Accounte Recelvable on December

31

(prior year) is

\$350,000

, Sales from Quarter

1

\$300,000

, Sales from Quarter

2

\$250,000

, Soles from Quarter

3

\$500,000

, and Sales from quarter

4

\$600,000

. Ueing budget standarde, the cash to be collected for the first quarter will consist of the accounte receivable balance on December

31

(prior year) and

70

\% of the sales made during the

1

st quarter. For future quartere,

70

\% of eales will be collected in each current quarter and

30

\% will be collected in the quarter after the cale.

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How much cash will Buffalo Corporation collect next year?

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1,930,000

\newline

2,000,000

\newline

1,470,000

\newline

1,820,000

Acircle with area

36 \pi

has a sector with a central angle of

48^{\circ}

. What is the area of the sector?

\newline

1

\newline

\frac{5}{24} \pi

\newline

\frac{1}{270} \pi

\newline

270 \pi

\newline

\frac{24}{5} \pi

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P=\frac{19}{2}+\frac{4}{25}d

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The amount of paint,

P

, in pints, that an art studio can buy with

d

dollars is given by the equation. What is the best interpretation of

\frac{4}{25}

as shown in the equation?

\newline

1

\newline

(A) The paint costs

\frac{4}{25}

of a dollar per pint

\newline

(B) One dollar can buy

\frac{4}{25}

of a pint of paint.

\newline

(C) The art studio started with

\frac{4}{25}

of a pint of paint.

\newline

(D) The art studio can purchase at most

\frac{4}{25}

of a pint of paint.

L=-5.6+1.1d

\newline

After tracking a species of migratory bird for ten years, the latitude,

L

, in degrees north for the species of bird

d

days after March

1

1

is approximated by the equation. By approximately how many degrees north does the latitude of the bird species change each day between March

1

1

\newline

1

\newline

-5.6

\newline

-1.1

\newline

1.1

\newline

5.6

The radius of a sphere is increasing at a rate of

2\,\text{mm/s}

. How fast is the volume increasing when the diameter is

40\,\text{mm}

A train traveling at a speed of

s

miles per hour applies its brakes before a buffer stop. Assuming

d \geq 0

s \geq 0

, the distance,

d

, in yards from the train to the buffer stop once the train comes to rest is:

\newline

d=0.5(-s^{2}-1.2s+184)

\newline

Which of the following equivalent expressions for

d

contains the traveling speed of the train, as a constant or coefficient, for which the train rests right at the buffer stop after applying its brakes?

\newline

1

\newline

-0.5s^{2}-0.6s+92.3

\newline

-0.5(s-13)(s+14.2)

\newline

-0.5(s+0.6)^{2}+92.48

\newline

(6.5-0.5s)(s+14.2)

A quarter is worth

\$0.25

and a half dollar is worth

\$0.50

\newline

a. A quarter has a diameter of

\frac{15}{16}

inch and a height of

\frac{1}{16}

inch. Find the surface area of a quarter. Round your answer to the nearest hundredth.

\newline

b. A half dollar has a diameter of

\frac{9}{8}

inches and a height of

\frac{3}{32}

inch. Find the surface area of a half dollar. Round your answer to the nearest hundredth.

Ahmed is modeling the area covered by a moss using the following equation:

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A=t^{2}+5.8 t+9.41

\newline

A

is the area covered in square centimeters, negative values of

t

represent a number of days before noon on July

1

2014

, and positive values of

t

represent a number of days after noon on July

1

2014

. Which of the following equivalent expressions contains the number of days before noon on July

1

2014

, as a constant or coefficient, when the moss started with its smallest area of

1

square centimeter?

\newline

1

\newline

(t-(-2.9))^{2}+1

\newline

(t-(-1))^{2}+3.8 t+8.41

\newline

(t-(-3.9))(t-(-1.9))+2

\newline

t(t-(-5.8))+9.41

Skyler climbs a watchtower to guard against forest fires.

\newline

R(h)=\sqrt{13h}

gives the distance, in kilometers, that Skyler can see when their eyes are

h

meters above the ground.

\newline

A(d)=\pi d^{2}

gives the area, in square kilometers, that Skyler can guard when they can see a distance of

d

\newline

Which expression models the area Skyler can guard when their eyes are

h

meters above the ground?

\newline

1

\newline

13\pi h

\newline

13\pi h^{2}

\newline

\sqrt{13\pi h^{2}}

\newline

169\pi h^{2}

After the outbreak of a mysterious disease, the average wait time at a hospital increased by

70\%

. Before the outbreak, the average wait time at the hospital was

m

\newline

Which of the following expressions could represent the average wait time at the hospital after the outbreak?

\newline

2

\newline

0.7m

\newline

\frac{170}{100}m

\newline

m+70

\newline

m+0.7

\newline

1.7m

The work done by stretching a certain spring increases by

0.13 x

joules per centimeter (where

x

is the displacement, in centimeters, beyond the spring's natural length).

\newline

How much work (in Joules) must be done in order to stretch the spring from

x=4

x=10

\newline

1

\newline

5

46

\newline

7

54

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\mathbf{1 0 . 9 2}

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\mathbf{1 5 . 0 8}