Question

Scenario: Two people in a major city with a population of 450,000 get infected with a rapidly spreading virus. If each day the infected person infects 2 additional people, do the following:
1) Create a table with two columns. The first column should be labeled as "Number of Days" and the second column "TOTAL Number People Infected." Fill in the table for days 0 through 5.
2) Using the data from the table created in step 1, create a graph with "Number of Days" on the x-axis and "Total Infected People" on the y- axis.
3) Create a mathematics model (the equation) for this scenario.
4) How long will it take the ENTIRE city to be infected (Round to 2 decimal) places?

Answer 1

1) See attachment.

2) See attachment

3) $y=2^{x+1}$

4) 17.8 days

Step-by-step explanation:

1)

The table is in attachment.

In this problem, we are told that the initial number of people infected ad day zero is two, so the first row is (0,2).

Then, we are told that each day, an infected person infects 2 additional people. Therefore, at day 1, the number of infected people will be 2*2=4.

Then, each of the 4 persons infect 2 additional persons, so the number of infected people at day 2 will be 4*2=8.

Continuing the sequence, the following days the number of infected people will be:

8*2 = 16

16*2 = 32

32*2 = 64

2)

The graph representing the situation is shown in attachment.

On the x-axis, we have represented the day, from zero to 5.

On the y-axis, we have represented the number of infected people.

We see that the points on the graph are:

0, 2

1, 4

2, 8

3, 16

4, 32

5, 64

3)

Here we have to create a mathematics model (so, an equation) representing this scenario.

First of all, we notice that the number of infected people at day 0 is 2:

$p(0)=2$

To write an equation, we call $x$ the number of the day; this means that at x = 0, the value of y (number of infected people) is 2:

$y=2$

Then, at day 1 (x=1), the number of infected people is doubled:

$y(1)=2y(0)=2\cdot 2 = 4$

And so on. This means that for each increase of x of 1 unit, the value of y doubles: so, we can represents the model as

$y=2\cdot 2^x$

Or

$y=2^{x+1}$

4)

Here we are told that the entire city has a population of

p = 450,000

people.

In order for the virus to infect the whole population, it means that the value of y must be equal to the total population:

y = 450,000

Substituting into the equation of the model, this means that

$450,000 = 2^{x+1}$

And solving for x, we find the number of days after which this will happen:

$log_2(450,000)=x+1\\x=log_2(450,000)-1=17.8 d$

So, after 17.8 days.