Chapter 18 Ratio, Proportion, & Variation Sect 18.1 : Ratio and Proportion A ratio conveys the notion of “relative magnitude”. Ratios are used.
Download ReportTranscript Chapter 18 Ratio, Proportion, & Variation Sect 18.1 : Ratio and Proportion A ratio conveys the notion of “relative magnitude”. Ratios are used.
Slide 1
Chapter 18
Ratio, Proportion, & Variation
1
Slide 2
Sect 18.1 : Ratio and Proportion
A ratio conveys the notion of “relative magnitude”.
Ratios are used to compare two quantities.
Remember the set of rational numbers?
2
Slide 3
A ratio can be used to compare quantities of the same kind
(can be expressed using the same units).
Example 1:
The TI-84 calculator display screen is 94 pixels by 62 pixels.
Write this as a ratio in simplest form.
3
Slide 4
Example 2:
Write as the ratio of whole numbers in simplest form.
5
3
5
to 2
1
10
4
Slide 5
Example 3:
Write the ratios in simplest form.
a) The ratio of 75 s to 3 min
b) The ratio of 2.7 kg to 60.5 g
5
Slide 6
A ratio can be used to compare denominate numbers with
different units. These ratios are also called rates.
Example
5 lb of fertilizer covers 1200 square feet. How many
square feet are covered per pound of fertilizer?
6
Slide 7
An equation stating that two ratios are equal is called a
________________________.
If
a
b
c
, then ad bc .
d
The product of the ______________ equals the product of the
________________.
7
Slide 8
Example 2:
To conserve energy and still allow for as much natural lighting as
possible, an architect suggest that the ratio of the area of a
window to the area of the total wall surface to be 5 to 12. Using
this ratio, determine the recommended area of a window to be
installed in a wall that measures 8 ft by 12 ft.
8
Slide 9
Examples from the book:
P. 494 # 26, 40, 44
9
Slide 10
10
Slide 11
Section 18.2: Variation
To estimate the number of phone calls expected per day
between two cities, one telecommunication company
used the following formula
C
0.02 P1 P2
d
2
,
which shows that the daily number of phone calls, C,
increases as the populations of the cities, P1 and P2, in
thousands, increase and decreases as the distance, d,
between the cities increases.
11
Slide 12
Variation
Variation formulas show how one quantity changes in
relation to other quantities.
Quantities can vary directly, inversely, or jointly.
12
Slide 13
Direct Variation
When you swim underwater, the pressure in your ears depends on
the depth at which you are swimming. The formula p 0.43 d
describes the water pressure, p, in pounds per square inch, at a
depth of d feet.
a) Find the water pressure in your ears at a depth of 10 ft.
b) Find the water pressure in your ears at a depth of 20 ft.
c) Find the water pressure in your ears at a depth of 40 ft.
13
Slide 14
Direct Variation
In this example, the water pressure is a constant multiple
of your underwater depth. If your depth is doubled, the
pressure is doubled; if the depth is tripled, the pressure is
tripled; etc.
The pressure in your ears is said to vary directly as your
underwater depth.
p 0.43 d
14
Slide 15
Direct Variation
When two quantities are in the same proportion and can be
expressed as an equation in the form y = k x, we say that
“y varies directly as x” or “y is directly proportional
to x”.
The (non-zero) number k is called the constant of
proportionality.
15
Slide 16
Direct Variation Example
The number of gallons of water, W, used when taking a shower varies
directly as the time, t, in minutes, in the shower. A shower lasting 5
minutes uses 30 gallons of water. How much water is used in a
shower lasting 11 minutes?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
16
Slide 17
Direct Variation
Direct variation can involve variables to higher powers.
y kx
n
We say that y is directly proportional to the nth power of x.
17
Slide 18
Direct Variation Example
The distance, s, that a body falls from rest varies directly as the
square of the time, t, of the fall. If skydivers fall 64 feet in 2
seconds, how far will they fall in 4.5 seconds?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
18
Slide 19
Inverse Variation
When two quantities are related by an equation in the
form
k
y
x
we say that “y varies inversely as x” or “y is inversely
proportional to x”.
The (non-zero) number k is called the constant of
variation.
19
Slide 20
Inverse Variation
For example, for a fixed distance, the time it takes to travel
that distance is inversely proportional to the rate at which one
drives.
The distance from Plattsburgh to Albany is 150 miles.
T im e
150
R ate
20
Slide 21
Inverse Variation Example
At a constant temperature, the pressure, P, of a gas container varies
inversely with the volume, V, of the container. The pressure of a gas
sample in a container whose volume is 8 cubic inches is 12 pounds per
square inch. If the sample expands to a volume of 22 cubic inches,
what is the new pressure of the gas?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
21
Slide 22
Combined Variation
In combined variation, direct and inverse
variation occur at the same time.
y
kx
y
22
Slide 23
Combined Variation Example
One’s intelligence quotient, or IQ, varies directly as a person’s mental
age and inversely as that person’s chronological age. A person with a
mental age of 25 and a chronological age of 20 has and IQ of 125. What
is the chronological age of a person with a mental age of 40 and an IQ
of 80?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
23
Slide 24
Joint Variation
Joint variation is a variation in which a variable varies
directly as the product of two or more other variables.
To show that y varies jointly as x and z, we would write
y kxz
24
Slide 25
Joint Variation
Newton’s famous formula for gravitation shows this:
F
G m1 m 2
d
2
The force, F, between two bodies varies jointly as the product
of their masses, m1 and m2, and inversely as the square of the
distance between them, d 2. (G is the gravitational constant.)
25
Slide 26
Joint Variation Example 1
The volume of a cone, V, varies jointly as its height, h, and the square
of its radius, r. A cone with a radius measuring 6 feet and a height
measuring 10 feet has a volume of 120 cubic feet. Find the volume of
a cone having radius of 12 feet and a height of 2 feet.
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
26
Slide 27
Joint Variation Example 2
The centrifugal force, C, of a body moving in a circle varies jointly with
the radius of the circular path, r, and the body’s mass, m, and inversely
with the square of the time, t, it takes to move about one full circle. A
6-gram body moving in a circle with radius 100 cm at a rate of 1 rev in
2 seconds has a centrifugal force of 6000 dynes. Find the centrifugal
force of an 18-gram body moving in a circle with radius 100 cm at a rate
of 1 revolution in 3 seconds.
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
27
Slide 28
End of Section
28
Chapter 18
Ratio, Proportion, & Variation
1
Slide 2
Sect 18.1 : Ratio and Proportion
A ratio conveys the notion of “relative magnitude”.
Ratios are used to compare two quantities.
Remember the set of rational numbers?
2
Slide 3
A ratio can be used to compare quantities of the same kind
(can be expressed using the same units).
Example 1:
The TI-84 calculator display screen is 94 pixels by 62 pixels.
Write this as a ratio in simplest form.
3
Slide 4
Example 2:
Write as the ratio of whole numbers in simplest form.
5
3
5
to 2
1
10
4
Slide 5
Example 3:
Write the ratios in simplest form.
a) The ratio of 75 s to 3 min
b) The ratio of 2.7 kg to 60.5 g
5
Slide 6
A ratio can be used to compare denominate numbers with
different units. These ratios are also called rates.
Example
5 lb of fertilizer covers 1200 square feet. How many
square feet are covered per pound of fertilizer?
6
Slide 7
An equation stating that two ratios are equal is called a
________________________.
If
a
b
c
, then ad bc .
d
The product of the ______________ equals the product of the
________________.
7
Slide 8
Example 2:
To conserve energy and still allow for as much natural lighting as
possible, an architect suggest that the ratio of the area of a
window to the area of the total wall surface to be 5 to 12. Using
this ratio, determine the recommended area of a window to be
installed in a wall that measures 8 ft by 12 ft.
8
Slide 9
Examples from the book:
P. 494 # 26, 40, 44
9
Slide 10
10
Slide 11
Section 18.2: Variation
To estimate the number of phone calls expected per day
between two cities, one telecommunication company
used the following formula
C
0.02 P1 P2
d
2
,
which shows that the daily number of phone calls, C,
increases as the populations of the cities, P1 and P2, in
thousands, increase and decreases as the distance, d,
between the cities increases.
11
Slide 12
Variation
Variation formulas show how one quantity changes in
relation to other quantities.
Quantities can vary directly, inversely, or jointly.
12
Slide 13
Direct Variation
When you swim underwater, the pressure in your ears depends on
the depth at which you are swimming. The formula p 0.43 d
describes the water pressure, p, in pounds per square inch, at a
depth of d feet.
a) Find the water pressure in your ears at a depth of 10 ft.
b) Find the water pressure in your ears at a depth of 20 ft.
c) Find the water pressure in your ears at a depth of 40 ft.
13
Slide 14
Direct Variation
In this example, the water pressure is a constant multiple
of your underwater depth. If your depth is doubled, the
pressure is doubled; if the depth is tripled, the pressure is
tripled; etc.
The pressure in your ears is said to vary directly as your
underwater depth.
p 0.43 d
14
Slide 15
Direct Variation
When two quantities are in the same proportion and can be
expressed as an equation in the form y = k x, we say that
“y varies directly as x” or “y is directly proportional
to x”.
The (non-zero) number k is called the constant of
proportionality.
15
Slide 16
Direct Variation Example
The number of gallons of water, W, used when taking a shower varies
directly as the time, t, in minutes, in the shower. A shower lasting 5
minutes uses 30 gallons of water. How much water is used in a
shower lasting 11 minutes?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
16
Slide 17
Direct Variation
Direct variation can involve variables to higher powers.
y kx
n
We say that y is directly proportional to the nth power of x.
17
Slide 18
Direct Variation Example
The distance, s, that a body falls from rest varies directly as the
square of the time, t, of the fall. If skydivers fall 64 feet in 2
seconds, how far will they fall in 4.5 seconds?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
18
Slide 19
Inverse Variation
When two quantities are related by an equation in the
form
k
y
x
we say that “y varies inversely as x” or “y is inversely
proportional to x”.
The (non-zero) number k is called the constant of
variation.
19
Slide 20
Inverse Variation
For example, for a fixed distance, the time it takes to travel
that distance is inversely proportional to the rate at which one
drives.
The distance from Plattsburgh to Albany is 150 miles.
T im e
150
R ate
20
Slide 21
Inverse Variation Example
At a constant temperature, the pressure, P, of a gas container varies
inversely with the volume, V, of the container. The pressure of a gas
sample in a container whose volume is 8 cubic inches is 12 pounds per
square inch. If the sample expands to a volume of 22 cubic inches,
what is the new pressure of the gas?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
21
Slide 22
Combined Variation
In combined variation, direct and inverse
variation occur at the same time.
y
kx
y
22
Slide 23
Combined Variation Example
One’s intelligence quotient, or IQ, varies directly as a person’s mental
age and inversely as that person’s chronological age. A person with a
mental age of 25 and a chronological age of 20 has and IQ of 125. What
is the chronological age of a person with a mental age of 40 and an IQ
of 80?
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
23
Slide 24
Joint Variation
Joint variation is a variation in which a variable varies
directly as the product of two or more other variables.
To show that y varies jointly as x and z, we would write
y kxz
24
Slide 25
Joint Variation
Newton’s famous formula for gravitation shows this:
F
G m1 m 2
d
2
The force, F, between two bodies varies jointly as the product
of their masses, m1 and m2, and inversely as the square of the
distance between them, d 2. (G is the gravitational constant.)
25
Slide 26
Joint Variation Example 1
The volume of a cone, V, varies jointly as its height, h, and the square
of its radius, r. A cone with a radius measuring 6 feet and a height
measuring 10 feet has a volume of 120 cubic feet. Find the volume of
a cone having radius of 12 feet and a height of 2 feet.
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
26
Slide 27
Joint Variation Example 2
The centrifugal force, C, of a body moving in a circle varies jointly with
the radius of the circular path, r, and the body’s mass, m, and inversely
with the square of the time, t, it takes to move about one full circle. A
6-gram body moving in a circle with radius 100 cm at a rate of 1 rev in
2 seconds has a centrifugal force of 6000 dynes. Find the centrifugal
force of an 18-gram body moving in a circle with radius 100 cm at a rate
of 1 revolution in 3 seconds.
a) Write the general equation:
b) Find the value of the constant of proportionality (k):
c) Write the specific equation:
d) Find the desired value:
27
Slide 28
End of Section
28