Section 3.5 – Mathematical Modeling
k
Inverse Variation - y 
Direct Variation - y  kx
x
x
2 4 6 8 10
y  kx 2 8 32 72 128 200
1
k2
y  kx 2
y  2  2  8
2
y  2  4   32
2
y  2  6   72
2
y  2  8   128
2
y  2 10   200
2
4
9
16
25
1
k
4
y  kx 2
1 2
y   2  1
4
1
2
y   4  4
4
1 2
y  6  9
4
1 2
 8   16
4
1
2
y  10   25
4
y
k
Inverse Variation - y 
Direct Variation - y  kx
x
x
2
4
6
8
10
k
5/4 5/16 5/36 5/64 1/20
y 2
x
5 5/4 5/9 5/16 1/50
k 5
k
y 2
x
5 5
5
5
y 2 
y 2 
2
4
8
64
5
5
5
1
y 2 
y 2 
4
16
10
20
5
5
y 2 
6
36
k  20
k
y 2
x
20
y 2 5
2
20 5
y 2 
4
4
20 5
y 2 
6
9
20 5

2
8
16
20
1
y 2 
10
50
y
Direct Variation - y  kx
x
y
y  kx
2  k 5 
2
k
5
2
y x
5
2
y  10 
5
y4
5
2
k
OR Inverse Variation - y 
x
10 15 20 25
4 6 8 10
10
y
x
10
y
10
y 1
k
y
x
k
2
5
k  10
NO
YES
DIRECT VARIATION
Direct Variation - y  kx
x
y
y  kx
24  k  5 
24
k
5
NO
k
OR Inverse Variation - y 
x
5 10 15 20 25
24 12 8 6 24/5
24
y
x
5
24
y
10 
5
y  48
120
y
x
120
y
10
y  12
k
y
x
k
24 
5
k  120
YES
INVERSE VARIATION
k
Inverse Variation - y 
x
x
Directly Proportional -  k
y
Direct Variation - y  kx
If x = 2 and y = 14, write a linear model that relates y to x if y is
directly proportional to x.
2
x
x 1
k  
k 
14
y
y 7
If x = 6 and y = 580, write a linear model that relates y to x if y is
directly proportional to x.
6
x
3
x
k  
k 
580
y 290
y
The simple interest (I) on an investment is directly proportional
to the amount of the investment (P). By investing $5000 in a
municipal bond, you obtained an interest payment of $187.50
after one year. Find a mathematical model that gives the
interest (I) for this municipal bond after one year in terms of the
amount invested (P).
I
k
P
187.50
k
5000
I 187.50

P
5000
The distance a spring is stretched (or compressed) varies
directly as the force on the spring. A force of 220 newtons
stretches a spring 0.12 meters. What force is required to
stretch the spring 0.16 meters?
D  kF
0.12  k  220 
0.12
k
220
0.12
D
F
220
0.12
0.16 
F
220
293.333  F
k
Inverse Variation - y 
x
x
Directly Proportional -  k
y
Direct Variation - y  kx
Write a mathematical model for each of the following:
A) y varies directly as the cube of x
y  kx3
B) h varies inversely as the square root of s
h
k
s
C) c is jointly proportional to the square of x and y3
c
k
2 3
x y
Write a mathematical model for each of the following. In each
case, determine the constant of proportionality.
A) y varies directly as the cube of x. (y = 81 when x = 3)
y  kx3
k 3
3
81  k  3 
B) h varies inversely as the square root of s. (h = 2 when s = 4)
k
k
h
2 2
k  32
4
s
C) c is jointly proportional to the square of x and y3
(c = 144 when x = 3 and y = 2)
144
c
k
k
k2
2
3
2 3
x y
3   2
k
Inverse Variation - y 
x
x
Directly Proportional -  k
y
Direct Variation - y  kx
The stopping distance d of an automobile is directly proportional
to the square of its speed s. A car required 75 feet to stop
when its speed was 30 mph. Estimate the stopping distance
if the brakes are applied when the car is traveling at 50 mph.
d
k
2
s
d
75
 2
2
s
30
75
k
2
30
d
75
 2
2
50
30
d  208.333