Question

At the beginning of the summer, the population of wasps is growing at a rate proportional to the population. From a population of 10 on May 1, the number of wasps grows to 50 in thirty days. If the growth continues to follow the same model, how many days after May 1 will the population reach 100?​

Answer 1

30 day after may 1 to june 1 will the population reach 100

Answer 2

In 50 days the population will reach 100.

Step-by-step explanation:

Let the rate of population be $x$. Let the initial population be $x_0$.

By the given problem,

$x \propto x_0$

let,

$x=x_0e^{kt}$                 (1)  (t=time, k=constant)

In thirty days the population increases from 10 to 50.

Therefore,

$x=40\\x_0=50-10=40\\t= 30$

Putting in (1) we get,

$40=10e^{30k}\\or, \ 4=e^{30k}\\\ln 4=30k\\k= \frac{\ln4}{30}$

Now, To find the time it will take to reach $x=100$, we will substitute the value of k in equation (1)

$100=10e^{\frac{\ln 4}{30}t}\\10=e^{\frac{\ln 4}{30}t}\\\ln 10=\frac{\ln 4}{30}t\\t= \frac{30\ln 10}{\ln 4}\\t=50$