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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