Question

The population of a colony of mosquitoes obeys the law of uninhibited growth. If there are 1000 mosquitoes initially and there are 1600 after 1​ day, what is the size of the colony after 4 ​days? How long is it until there are 10,000 ​mosquitoes?

Answer 1

Answer:

1) The size of the colony after 4 ​days is 6553 mosquitoes

2) After $t = 4.9\ days$

Step-by-step explanation:

To answer this question you must use the growth formula

$N = N_0e ^ {kt}$

Where

$N_0$ is the initial population of mosquitoes = 1000

t is the time in days

k is the growth rate

N is the population according to the number of days

We know that when t = 1 and $N_0=1000$ then N = 1600

Then we use these values to find k.

$1600 = 1000e ^{k(1)}\\\\\frac{1600}{1000} = e ^ k\\\\ln(\frac{1600}{1000}) = k\\\\k = 0.47$

Now that we know k we can find the size of the colony after 4 days.

$N = 1000e ^ {0.47(4)}\\\\N = 6553\ mosquitoes$

To know how long it should take for the population to reach 10,000 mosquitoes we must do N = 10000 and solve for t.

$10000 = 1000e ^ {0.47t}\\\\10 = e ^ {0.47t}\\\\ln(10) = 0.47t\\\\t = \frac{ln(10)}{0.47}\\\\t = 4.9\ days$

Answer 2

Answer:

The size of the mosquito colony after 4 ​days is about $6553$ and it will take about $4.9$ days to become $10,000$ mosquitoes.

Explanation:

Step 1 :

Given,

Initially $1000$ mosquitoes

After $1$ day the mosquitoes increases to $1600$.

After$4$ days=?

When it reaches about $10000$ mosquitoes?

Let,

To find the population of mosquitoes after$4$ days,

Using the formula,$N=N_{0} e^{k t}$

Here,

N=Number of population after $1$ day$(1600)$

$N_{0}=$Initial population$(1000)$

e is constant value

k=rate of growth of mosquitoes

t=growth time of mosquitoes$(1)$

Find $N_{1$,population after $4$ days

First,

Substituitng the values in the formula,

$1600=1000 e^{k(1)}$

k value,

$\frac{1600}{1000}=e^{k}$

Opposite side change of $e$,gives

$\ln \left(\frac{1600}{1000}\right)=k$

Cancelling zeros,

$\ln \left(\frac{16}{10}\right)=k$

Simplifying,

$\ln (16)-\ln (10)=k$

$k=0.47$

Apply the value of k in the formula,

$N_{1} =1000 e^{0.47(4)}$

$N_{1} =1000 e^{1.88}$

$N_{1} =6553$

The size of the colony after $4$ days is $6553$ mosquitoes.

Step 2:

Then,

Time taken to reach about $10000$ mosquitoes is

Here we know that the number of mosquitoes$=10000$

Time taken$=?$

Formula:

$N=N_{0} e^{k t}$

$N=10,000$

$N_{0}=1000$

$k=0.47$

By the values,

$10,000=1000 e^{0.47 t}$

$10,000/1000= e^{0.47 t}$

$10= e^{0.47 t}$

$\ln (10)=0.47 t$

$t=\frac{\ln (10)}{0.47}$

$t=4.9$

It takes about $4.9$ days to reach the population of $10,000$ mosquitoes.