Math-3

7-5

Modeling Density

and

Rewriting formulas for the variable of interest

What is happening?

Why?

How would you compare the two collections?

Devise a “rate” type quantity so that we can compare the

two amounts.

The boxes above have a side length of 2 inches.

3in 8

spheres 273in

spheres 3.375

3in 8

spheres 123in

spheres 1.5

number/unit volume

mass/unit volume = density

The total mass of steel used in the construction of a

car is 1800 lbm.

The density of steel is 490 pounds (mass) per cubic foot.

What is the volume of steel in a car?

vol

massdensity

vol

lbm 1800

ft

lbm 490

3

lbm 1800lbm 490

ft3 vol

vollbm 490

ft*lbm 1800 3

3ft 67.3vol

What does “surface area” mean?Surface area: The area of the surface of the shape.

Why would this information be important?

Helps you to know how much material you need to

build, paint, or cover the item.

Vocabulary

Formula: an equation that shows the relationship between two or more quantities.

Examples of formulas you’ve seen are:

2 rA

hwLV **

rhAcylinder 2)r (2 2

rC 2

The surface area of a cylinder is….?8 inches

10 inches

Net a the “flattened” version of

a 3-dimensional shape.

sides

Bottom

Top

The surface area of a cylinder is….?8 inches

10 inches

The surface area of a

cylinder is made up of 2

circles and 1 rectangle.

sides

Bottom

Top

8 inches

10 inches

Top

Bottom

2

circle r a rea

wlrea *a rectangle

topof ncecircumfere*heightarea rectangle

)2(*a rectangle rhrea )4*2(*10 inin

2(4in) 2

circle in 16a rea

2in 80π222

cylinder in 16in 80in 16area Surface 2

cylinder in 112area Surface

What is the “surface area” of the prism?

10 in

3 in

Net a the “flattened” version of

a 3-dimensional shape.

3 in“h” in4 in

10 in

You can think of the “lateral sides” as 3

surfaces OR you can think of it as the

rectangular portion of the net.

The surface area of a pyramid is….?

The sum of the area of the faces.

Rectangular Pyramid has a 4-sided

base: it has four triangular faces.

The “slant height” of the pyramid is

the “height” of the triangular face.

The surface area of a rectangular pyramid is 1

rectangle and 4 triangles.

4 in

6 inches

The sum of the area of the 5 faces.

4 in

2

base in 16* area wl

heightslant *2

1 areaface base

2 in

6 inches

22 )6()2(heightslant inin

2in 40heightslant in 3.6

in 3.6*in 4*2

1 area face

2

face in 6.12 area

22

total in 16)in 6.12(4 area

2

total in 4.66 area

h)area base (3

1 volume pyramidr rectangula

h*base) area( volumecylinder

h*)base area(3

1 volumecone

h*base) of (area volumeprism

2

sphere r 4area surf.

3

sphere r 4*3

1 volume

What does “f(x)” mean?

Rule name Input variable

It means that there is a rule, named “f” whose output is

a result of “doing math” on the input to the variable ‘x’.

Example: 32)( xxf

‘x’ is a place-holder in the rule where we

substitute in the input value.

Fill in the blank for each function: “______” is a function of ____”

“g is a function of n”32)( nng

tetA 02.010)(

510016)( 2 ttth

225)( xxk

5)2()4( 22 yx

“h is a function of t”

“k is a function of x”

“A is a function of t”

neither

2)4(252 xy

2)2(254 yx

“y is a function of x”

“x is a function of y”

Rewriting formulas

We say that one quantity is a function of one or more other quantities.

2 rA

wLA *

rC 2

),( wLfA

),( hrfA

) ( rfA

) ( rfC

rhAcylinder 2)r (2 2

Rewriting formulasFor the area formula, write length as a function of Area and width.

wLA * ),( wLfA

w

AL

This means “solve” for length in the formula.

Using the property of equality, divide left/right of the “=“ sign by “A”

),( wAfL

Rewriting formulasRewrite the circumference formula as radius as a function of circumference.

rC 2 ) ( rfC

) ( Cfr 2

Cr

Rewriting formulas

Rewrite the formula so that it is in the form:

)(FfC

)(CfF

)32(9

5 FC

325

9 FC

325

9 CF

)32(9

5

5

9

5

9

FC

Describe the transformation of the parent function:

5)4(3 2 xyReflected across x-axis, VSF = 3, left 4, down 5

Solve the equation for ‘x’.

5)4(3 2 xy2)4(35 xy

2)4(3

5

x

y

2)4(3

5

x

y

43

5

x

y

3

54

yx

05 : yR

yR 5 :

5y : R

Solve the equation for ‘x’

3)1(5 2 xy

2)1(53 xy

2)1(5

3

x

y

15

3

x

y

3

31

yx

03 : yR

3 : yR

Solve the equation for ‘x’

4)2log(6 xy

)2log(64 xy

)2log(6

4

x

y

210 6

4

x

y

6

4

102

y

x

Rewriting formulas

Rewrite the formula as:

2

)(* 21 bb

hA

),b, ( 21 bhfA

)b,A, ( 21 hfb

2b

1b

h

)(2 21 bbhA

21

2bb

h

A

21

2b

h

Ab

212 hbhbA

122 hbhbA

h

hbAb 2

1

2

Are the two formulas are equivalent?

Expressions from PhrasesWhat is a mathematical expression that represents the following?

Three more than twice a number

Five less than three times a number

The width is 4 times the length.

The area of a rectangle whose width is

4 times its length.

32 x

53 x

Lw 4

LwA

)4( LLA

Expressions from PhrasesWhat is a mathematical expression that represents the following?

The width of a rectangle is 3 less than

twice its length.

32 Lw

Solve a totally non-recognizable quadratic equation by graphing.

)2)(3(20 xx

)2)(3(

20

xxy

y

Finding the dimensions of a rectangle.The length of one side of a rectangle is three more than

two times a number. What is the expression for the

length of the side?

x - 2

2x + 3

The width of the

rectangle is two less than

the number. What is the

expression for the length

of the side?

If the area of the rectangle is 400 square inches,

what is the length and width of the rectangle?

Finding the dimensions of a rectangle.

x - 2

2x + 3

A = 400, length = ? Width = ?

WLA * 32 xL 2 xw

By substitution:

WL*400

By substitution:

Wx *)32(400

By substitution:

)2)(32(400 xx

400A

)2)(32(400 xx

32 xL2 xw

3)5.14(2 L

32L

25.14 w

5.12w

)5.12)(32(400

check

)2)(32(

400

xxy

y

Solve by graphing

system of equations.

14x 5.14x

Do both values of ‘x’ give you an

area that is a positive number?

Expressions from PhrasesWhat is a mathematical expression that represents the following?

The length of a side of a rectangle that

has been reduced on each end by the

same number.

xL 2

Maximizing Volume of a Box

x

9”

Corners are cut out of a 9” x 21” piece of cardboard.

Then the sides are folded up along the dotted lines.

xx

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

x

x

9”

x

x

21”

x

x

x

xWhen the sides arefolded up, what willbe the length of the bottom of the box (function of “x”) ?

?

Maximizing Volume of a Box

x

9”

Corners are cut out of a 9” x 21” piece of cardboard.

Then the sides are folded up along the dotted lines.

xx

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

x

x

9”

x

x

21”

x

x

x

xWhen the sides arefolded up, what willbe the length of the bottom of the box (function of “x”) ?

21 – 2x

Maximizing Volume of a Box

x

9”

Corners are cut out of a 9” x 21” piece of cardboard.

Then the sides are folded up along the dotted lines.

xx

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

x

x

9”

x

x

21”

x

x

x

x

21 – 2x

When the sides arefolded up, what willbe the length of the side of the box (function of “x”) ?

?

Maximizing Volume of a Box

x

9”

Corners are cut out of a 9” x 21” piece of cardboard.

Then the sides are folded up along the dotted lines.

xx

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

x

x

9”

x

x

21”

x

x

x

x

21 – 2x

When the sides arefolded up, what willbe the length of the side of the box (function of “x”) ?

9 – 2x

Maximizing Volume of a Box

x

9”

Corners are cut out of a 9” x 21” piece of cardboard.

Then the sides are folded up along the dotted lines.

xx

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

x

x

9”

x

x

21”

x

x

x

x

21 – 2x

9 – 2xWhat will be the height of the box?

x

Maximizing Volume of a Box

x

9”

Write an equation that models the volume of the box as a

function the height of the box: Vol(x) = ?

xx

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

9”

x

x

x

9”

x

x

21”

x

x

x

x

21 – 2x

9 – 2x

V(x) = L*w*h

V(x) = (21-2x)(9-2x)(x)

V(x) = L*w*h

V(x) = (21-2x)(9-2x)(x)

What is the “implied domain”

of this problem?

9-2x

21-2x

9 - 2x = 0

x = 4.5”

What is the “maximum possible volume of a box made

from this piece of cardboard?

Stop here

“Proportional to….”

The distance you travel is proportional to the speed at

which you travel.

The distance you travel is proportional to the time during

which you travel.

The money that your earn is proportional to your hourly

wage.

The money that you earn is proportional to the amount

of time that you work. time $

speed α distance

time distance

wage $

The “constant of proportionality”

The distance you travel is proportional to the

speed at which you travel.

speed α distance

speed *K distance

We call “K” the constant of proportionality.

Distance is a function of speed.

Inverse ProportionalityThe longer you go without food, your weight goes down.

time

1 eight w

time

Kweight

Time without food goes up, weight goes down.

Time without food is inversely proportional to weight.

2)(d

kdF

The force of gravity depends upon the gravitational

constant (a unit conversion factor), the mass of the

two bodies that are attracting each other and the

distance between them.

2

21

d

mgmF

Gravitation force is inversely proportional to the

“square” of the distance.

baxxf )(

The “ideal gas law” states that the relationship between the pressure in an enclosed gas (like a balloon) is a function of the temperature of the gas.

PkV

The volume of an enclosed gas is inversely proportional to the pressure (if temperature is constant).

nRTPV P

nRTV

Can you give some examples of real world quantities that are powers?

2)( xxf

2)( KrrA

Area of a circle is proportional to the square of its radius.

2πrA

If we double the radius, by what factor does the area

change?2π(2r)newA

2πr4newA

oldnew AA 4

3)( xxf

Can you give some examples of real world quantities that are powers?

Volume of a cube is proportional to the cube of its side length.

3)( KssV 3sV

If we triple the side length, by what factor does the

volume change?

3)3( sVnew 327sVnew

oldnew VolV *27