Question

6.) a new head cold virus is being spread slowly around the world. in 2012, a household of 2 people both had the virus. in 2013, 50 people had been infected so far. in 2014, 1250 people had been infected so far. a.) write an equation for the situation, using n for total number of people infected, and t for time in years since 2012. b.) use your equation to predict how many total people would have been infected by 2018.

Explanation:

Step1: Identify growth type, set model

This is exponential growth, so use $n(t) = n_0 \cdot r^t$. In 2012 ($t=0$), $n_0=2$.

Step2: Solve for growth rate $r$

Use 2013 data ($t=1$, $n=50$):
$$50 = 2 \cdot r^1$$
$$r = \frac{50}{2} = 25$$
Confirm with 2014 ($t=2$): $2 \cdot 25^2 = 2 \cdot 625 = 1250$, which matches. The equation is $n(t) = 2 \cdot 25^t$.

Step3: Calculate $t$ for 2018

2018 is $2018-2012=6$ years after 2012, so $t=6$.

Step4: Compute 2018 infections

Substitute $t=6$ into the equation:
$$n(6) = 2 \cdot 25^6$$
$$25^6 = (5^2)^6 = 5^{12} = 244140625$$
$$n(6) = 2 \cdot 244140625 = 488281250$$

Answer:

a.) $n(t) = 2 \cdot 25^t$
b.) $488281250$