Question

A species of beetles grows 32% every year. Suppose 100 beetles are released into a field. How many beetles will there be in 10 years? A species of beetles grows 32% every year. Suppose 100 beetles are released into a field.
How many beetles will there be in 20 years? A species of beetles grows 32% every year. Suppose 100 beetles are released into a field.
About when will there be 100,000 beetles?

Answer 1

Given that a species of beetles grows 32% every year.

So growth rate is given by

r=32%= 0.32

Given that 100 beetles are released into a field.

So that means initial number of beetles P=100

Now we have to find about how many beetles will there be in 10 years.

To find that we need to setup growth formula which is given by

$A=P(1+r)^n$ where A is number of beetles at any year n.

Plug the given values into above formula we get:

$A=100(1+0.32)^n$

$A=100(1.32)^n$

now plug n=10 years

$A=100(1.32)^{10}=100(16.0597696605)=1605.97696605$

Hence answer is approx 1606 beetles will be there after 10 years.

To find answer for 20 years plug n=20 years

$A=100(1.32)^{20}=100(257.916201549)=25791.6201549$

Hence answer is approx 25791 beetles will be there after 20 years.

Now we have to find time for 100000 beetles so plug A=100000

$A=100(1.32)^n$

$100000=100(1.32)^n$

$1000=(1.32)^n$

$log(1000)=n*log(1.32)$

$\frac{\log\left(1000\right)}{\log\left(1.32\right)}=n$24.8810001465=n

Hence answer is approx 25 years.

Answer 2

Answer:

$P_{10} = 1605$ beetles

 $P_{20} = 25791$  beetles

In 24.88 years there will be 100 000 beetles

Step-by-step explanation:

Let's call $P_t$ the beetle population that is in year t. If t starts in year 0, with 100 beetles, and the population grows 32% each year, then the population of beetles that will occur the following year is:

$P_1 = P_0 + 0.32P_0$

If we write this equation for a year t, then $P_t$ will have the following form:

$P_t = P_0 (1 + 0.32) ^ t$

Now we find $P_t = 10$

$P_{10} = 100 (1 + 0.32) ^ {10}$

$P_{10} = 1605$ beetles

In 20 years there will be:

$P_{20} = 100 (1 + 0.32) ^ {20}$

$P_{20} = 25791$  beetles

To know when there will be 100 000 beetles we equal Pt to 100 000 and we clear t.

$100000 = 100 (1 + 0.32) ^ t\\\\ ln (1000) = t * ln (1 + 0.32)\\\\ t = \frac{ln (1000)}{ln (1 + 0.32)}$

t = 24.88 years

In 24.88 years there will be 100 000 beetles