section 3.5 – mathematical modeling. direct variation - inverse variation - x 2 4 6 8 10 8 32 72...

Section 3.5 – Mathematical Modeling

Direct Variation - y kx Inverse Variation - k

yx

2y kxx 2 4 6 8 10

k 22y kx

2y 2 2 8

2y 2 4 32

2y 2 6 72

2y 2 8 128

2y 2 10 200

8 32 72 128 200

1k

4

2y kx

21y 2 1

4

21y 4 4

4

21y 6 9

4

21y 8 16

4

21y 10 25

4

1 4 9 16 25


yx

2

ky

x

x 2 4 6 8 10

k 5

2

ky

x

2

5 5y

2 4

2

5 5y

4 16

2

5 5y

6 36

2

5 5y

8 64

2

5 1y

10 20

5/4 5/16 5/36 5/64 1/20

k 20

2

ky

x

2

20y 5

2

2

20 5y

4 4

2

20 5y

6 9

2

20 5y

8 16

2

20 1y

10 50

5 5/4 5/9 5/16 1/50


yx

x 5 10 15 20 25y 2 4 6 8 10

OR

y kx

2 k 52

k5

2y x

5

2y 10

5

y 4

ky

x

k2

5

k 10

10y

x

YES

10y

10

y 1

NO

DIRECT VARIATION


yx

x 5 10 15 20 25y 24 12 8 6 24/5

OR

y kx

24 k 524

k5

24y x

5

24y 10

5

y 48

ky

x

k24

5

k 120

120y

x

NO

120y

10

y 12

YES

INVERSE VARIATION

Directly Proportional -


yx

x

ky

If x = 2 and y = 14, write a linear model that relates y to x if y isdirectly proportional to x.

xk

y

2k

14

If x = 6 and y = 580, write a linear model that relates y to x if y isdirectly proportional to x.

xk

y

x6 3

y 2k

058 90

x 1

y 7

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 $187.50after one year. Find a mathematical model that gives the interest (I) for this municipal bond after one year in terms of theamount invested (P).

I

Pk

5

1

0

87.50

00k

I 187.50

P 5000

The distance a spring is stretched (or compressed) variesdirectly as the force on the spring. A force of 220 newtonsstretches a spring 0.12 meters. What force is required to stretch the spring 0.16 meters?

D Fk

00.12 k 22

2

0.12

20k

0.12FD

220

0.120.16 F

220

293.333 F

Write a mathematical model for each of the following:

A) y varies directly as the cube of x

B) h varies inversely as the square root of s

C) c is jointly proportional to the square of x and 3y



yx

x

ky

3y kx

kh

s

2 3

c

x yk

Write a mathematical model for each of the following. In eachcase, determine the constant of proportionality.

A) y varies directly as the cube of x. (y = 81 when x = 3)

B) h varies inversely as the square root of s. (h = 2 when s = 4)

C) c is jointly proportional to the square of x and 3y

3y kx

kh

s

2 3

ck

x y

381 k 3

k 3

22

k

4 k 32

(c = 144 when x = 3 and y = 2)

2 3

144

3 2k k 2

The stopping distance d of an automobile is directly proportionalto the square of its speed s. A car required 75 feet to stopwhen its speed was 30 mph. Estimate the stopping distanceif the brakes are applied when the car is traveling at 50 mph.



yx

x

ky

2

d

sk

2

75

30k

2 2

7

s

d 5

30

2 2

7

50

d 5

30

d 208.333

section 3.5 – mathematical modeling. direct variation - inverse variation - x 2 4 6 8 10 8 32 72...

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