# To express: The given variation as equation.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 1.11, Problem 48E

(a)

To determine

## To express: The given variation as equation.

Expert Solution

The equation of given variation is r=k(x(5000x)) .

### Explanation of Solution

Jointly proportional:

Consider the equation,

z=kxy .

Here, z is proportional to product of xy or z is jointly proportional to x and y both statements are same.

k is constant of proportionality.

The rate r at which a disease spreads in population of size P is jointly proportional to the number of people infected and number of people who are not affected.

Mathematically,

r=k(x(Px))

Substitute 5000 for P in above equation.

r=k(x(5000x)) (1)

Here,

r is the rate at which disease spread, x is the number of people infected, 5000 is population of the town.

Thus, the equation of the given variation is r=k(x(5000x)) .

(b)

To determine

### To compare: The rate of spread of the infection in the two given cases.

Expert Solution

The rate of spread of infection in case 2 as compared to case 1 is 80 times.

### Explanation of Solution

Given:

Case 1:

Only 10 out of 5000 peoples are infected.

Case 2:

1000 out of 5000 people are infected.

Calculation:

To calculate, rate of spread of infection (r1) in case 1, substitute 10 for x in equation (1).

r=k(x(5000x))r1=k(10(500010))r1=k(10(4990))r1=k(49900) (2)

To calculate, rate of spread of infection (r2) in case 1, substitute 1000 for x in equation (1).

r=k(x(5000x))r2=k(1000(50001000))r2=k(1000(4000))r2=k(4000000) (3)

Divide equation (2) by equation (3).

r1r2=k(49900)k(4000000)r1r2=.0124

Simplify above equation.

r2=80.16r1

Thus, the rate of spread of infection when 1000 people are infected as compared to when 10 people are infected is approximately 80 times.

(c)

To determine

### To find: The rate of spread of infection when entire population is infected.

Expert Solution

The rate of spread of infection is zero.

### Explanation of Solution

Calculation:

If all the population is infected the value of x is 5000 , substitute 5000 for x in equation (1).

r=k(x(5000x))r=k(5000(50005000))r=0

Thus, rate of spread of infection is zero.

This answer makes intuitive sense because if all the peoples are infected there is no one left to be infected. Hence rate of spread of infection is zero.

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