4.6 Direct Variation
Notation
•
•
•
•
•
•
Y varies directly as X
y=kx
k is constant of variation
Y is proportional to X
y=kx
As one goes up, the other goes up.
Example 1
• Distance varies directly as time. A car travels
100 miles at a constant speed in 2 hours.
• D=kt
• 100=k(2)
• k=50
• D=50t
Example 2
• The charge(in dollars) to
customers for electricity(in
Kilowatt-hours) varies directly
as the number of kilowatthours used. It costs $52 to
use 800 kilowatt-hours. How
much to use 1000 kilowatthours?
C  kE
52  k (800)
52
13
k
800
k
200
C 
13
E
200
C 
13
(1000)
200
C  $65
y  kx
n
The distance a body falls from rest varies directly as
the square of the time it falls. If a skydiver falls 64ft in
2 sec, how far will she fall in 8 sec?
D  kt
2
64  k (2)
D  16 t
2
2
D  16(8)
2
64  4k
D  16(64)
16  k
D  1024 ft
Joint Variation
• Y varies jointly as X and Z
• y=kxz
• The interest on a loan is given by I=prt. Here, for a
given principal p, the interest earned I varies jointly as
the interest rate r, and the time t. If an investment earns
$100 interest at 5% for 2 yr, how much interest would
the same principal earn at 4.5% for 3 yr?
I  prt
100  p (.05)(2)
100  .1 p
1000  p
I  1 0 0 0 rt
I  1000(.045)(3)
I  $135
Homework
Pg.235:9,10,13,14,21-24,29,30
Inverse Variation
y 
k
o r xy  k
x
• As one variable increases, the other decreases.
• Y varies inversely as X
Example 1
• Volume of gas varies inversely as the pressure.
For a certain gas, the volume is 10 cm cubed when
the pressure is 6 kg per cm squared. Find the
volume when pressure is 12kg per cm squared.
V 
10 
k
P
k
6
k  60
V 
V 
60
P
60
12
V  5 cm
3
Combined Variation
• Body mass index (BMI) varies directly as a person’s weight in pounds and
inversely as the square of a person’s height in inches. A person who is
118lbs and has a height of 64 inches has a BMI of 20. Find the BMI of a
person who has weight of 165lbs and height of 70inches.
B 
20 
k  694
kw
2
h
k (118)
(64)
h
2
(64) * 20  118k
2
2
k 
B 
(64 ) * 20
118
694 w
B 
2
6 9 4 (1 6 5)
70
2
B  23
Homework
Pg. 235: 1-8,11,12,25-28,31-34