Question

the flu is starting to hit fairfax. currently, there are 647 people infected, and that number is growing at a rate of 13% per day. overall, how many people will have gotten the flu in 5 days? if necessary, round your answer to the nearest whole number. people

Explanation:

Step1: Identify the formula for exponential growth

The formula for exponential growth is \( A = P(1 + r)^t \), where \( P \) is the initial amount, \( r \) is the growth rate (in decimal), \( t \) is the time, and \( A \) is the amount after time \( t \).
Here, \( P = 647 \), \( r = 0.13 \) (since 13% = 0.13), and \( t = 5 \).

Step2: Substitute the values into the formula

Substitute \( P = 647 \), \( r = 0.13 \), and \( t = 5 \) into the formula:
\( A = 647(1 + 0.13)^5 \)
First, calculate \( 1 + 0.13 = 1.13 \).
Then, calculate \( 1.13^5 \). Let's compute that:
\( 1.13^2 = 1.2769 \)
\( 1.13^3 = 1.2769 \times 1.13 = 1.442897 \)
\( 1.13^4 = 1.442897 \times 1.13 = 1.63047361 \)
\( 1.13^5 = 1.63047361 \times 1.13 \approx 1.8424351793 \)

Step3: Multiply by the initial amount

Now, multiply \( 647 \) by \( 1.8424351793 \):
\( 647 \times 1.8424351793 \approx 647 \times 1.8424 \)
Calculate \( 647 \times 1.8424 \):
\( 600 \times 1.8424 = 1105.44 \)
\( 47 \times 1.8424 = 86.5928 \)
Add them together: \( 1105.44 + 86.5928 = 1192.0328 \)
Rounding to the nearest whole number, we get \( 1192 \). Wait, but let's do the multiplication more accurately:
\( 647 \times 1.8424351793 = 647 \times 1.8424351793 \)
Calculate \( 1.8424351793 \times 647 \):
\( 1.8424351793 \times 600 = 1105.46110758 \)
\( 1.8424351793 \times 40 = 73.697407172 \)
\( 1.8424351793 \times 7 = 12.8970462551 \)
Add them: \( 1105.46110758 + 73.697407172 = 1179.15851475 + 12.8970462551 = 1192.05556101 \)
So, rounding to the nearest whole number, it's \( 1192 \)? Wait, no, wait, maybe I made a mistake in the exponent calculation. Wait, let's use a calculator for \( 1.13^5 \):
\( 1.13^5 = e^{5 \ln(1.13)} \approx e^{5 \times 0.122276394} \approx e^{0.61138197} \approx 1.842435 \)
Then \( 647 \times 1.842435 = 647 \times 1.842435 \)
Let's do \( 647 \times 1.842435 \):
\( 647 \times 1 = 647 \)
\( 647 \times 0.8 = 517.6 \)
\( 647 \times 0.04 = 25.88 \)
\( 647 \times 0.002435 = 647 \times 0.002 + 647 \times 0.000435 = 1.294 + 0.281445 = 1.575445 \)
Add them: \( 647 + 517.6 = 1164.6 + 25.88 = 1190.48 + 1.575445 = 1192.055445 \)
So, rounding to the nearest whole number, it's \( 1192 \). Wait, but let's check with a calculator:
\( 647*(1.13)^5 \)
First, \( 1.13^5 = 1.13*1.13=1.2769; 1.2769*1.13=1.442897; 1.442897*1.13=1.63047361; 1.63047361*1.13=1.8424351793 \)
Then \( 647*1.8424351793 = 647*1.8424351793 \approx 1192.055 \), so rounded to the nearest whole number is \( 1192 \). Wait, but maybe I miscalculated. Wait, let's do 647*1.842435:

647 * 1.842435:

1.842435 * 600 = 1105.461

1.842435 * 40 = 73.6974

1.842435 * 7 = 12.897045

Sum: 1105.461 + 73.6974 = 1179.1584 + 12.897045 = 1192.055445, so yes, 1192.

Wait, but let's check with another approach. Maybe the formula is correct. The initial number is 647, growing at 13% per day, so each day it's 1.13 times the previous day. So after 1 day: 647*1.13 = 731.11

After 2 days: 731.11*1.13 = 826.1543

After 3 days: 826.1543*1.13 = 933.554359

After 4 days: 933.554359*1.13 = 1054.91642567

After 5 days: 1054.91642567*1.13 = 1192.05556091

Ah, there we go. So after 5 days, it's approximately 1192.055, which rounds to 1192. Wait, but wait, the problem says "overall, how many people will have gotten the flu in 5 days". Wait, is it the total number, or the number on day 5? Wait, the problem says "currently, there are 647 people infected, and that number is growing at a rate of 13% per day. Overall, how many people will have gotten the flu in 5 days?" Wait, maybe it's the total number, but actually, in exponential growth, the formula \( A = P(1 + r)^t \) gives the number of infected people on day \( t \), assuming that the growth is continuous or daily compounded. So if we start with 647, and each day the number grows by 13%, then after 5 days, the number of infected people is \( 647*(1.13)^5 \approx 1192 \). So that's the number of people who have gotten the flu by day 5 (including the initial 647? Wait, no, the initial 647 are already infected. Wait, the problem says "currently, there are 647 people infected, and that number is growing at a rate of 13% per day. Overall, how many people will have gotten the flu in 5 days?" Wait, maybe it's the total number of people who have gotten the flu by day 5, which would be the number of infected people on day 5, because each day the new infections are added to the previous. So the formula is correct. So the calculation is correct.

Answer:

1192