Answered

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 :

lublana

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

[tex]A=P(1+r)^n[/tex] where A is number of beetles at any year n.

Plug the given values into above formula we get:

[tex]A=100(1+0.32)^n[/tex]

[tex]A=100(1.32)^n[/tex]



now plug n=10 years

[tex]A=100(1.32)^{10}=100(16.0597696605)=1605.97696605[/tex]


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


To find answer for 20 years plug n=20 years

[tex]A=100(1.32)^{20}=100(257.916201549)=25791.6201549[/tex]


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

[tex]A=100(1.32)^n[/tex]

[tex]100000=100(1.32)^n[/tex]

[tex]1000=(1.32)^n[/tex]

[tex]log(1000)=n*log(1.32)[/tex]

[tex]\frac{\log\left(1000\right)}{\log\left(1.32\right)}=n[/tex]24.8810001465=n


Hence answer is approx 25 years.

carlosego

Answer:

[tex]P_{10} = 1605[/tex] beetles

 [tex]P_{20} = 25791[/tex]  beetles

In 24.88 years there will be 100 000 beetles

Step-by-step explanation:

Let's call [tex]P_t[/tex] 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:

[tex]P_1 = P_0 + 0.32P_0[/tex]

If we write this equation for a year t, then [tex]P_t[/tex] will have the following form:

[tex]P_t = P_0 (1 + 0.32) ^ t[/tex]

Now we find [tex]P_t = 10[/tex]

[tex]P_{10} = 100 (1 + 0.32) ^ {10}[/tex]

[tex]P_{10} = 1605[/tex] beetles

In 20 years there will be:

[tex]P_{20} = 100 (1 + 0.32) ^ {20}[/tex]

[tex]P_{20} = 25791[/tex]  beetles


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

[tex]100000 = 100 (1 + 0.32) ^ t\\\\ ln (1000) = t * ln (1 + 0.32)\\\\ t = \frac{ln (1000)}{ln (1 + 0.32)}[/tex]

t = 24.88 years

In 24.88 years there will be 100 000 beetles