Question

A population of 550 rabbits is increasing by 2.5% each year the function y = 5.5 (1.025)× give the population of rabbits in hundreds X year from now about how long will it take the population to reach 600 rabbits? 1200 rabbits?

Answer 1

Given function : y = 5.5 (1.025)×, Where exponent x is number of years from now and y give the population of rabbits in hundreds.

We need to find the time(s) when there will be 600 rabbits and 1200 rabbits.

Solution: We know, y represents the population of rabbits in hundreds.

a) Plugging y=600 in given function, we get

600 = 5.5 (1.025)×

Dividing both sides by 5.5

600/5.5 = 5.5 (1.025)× / 5.5

600/5.5 = (1.025)×

Taking natrual log on both sides, we get

ln (600/5.5) = ln (1.025)×

ln (600/5.5) = x ln (1.025)

Dividing both sides by ln(1.05), we get

$\frac{ln\frac{600}{5.5}}{ln(1.05)}=x$

x= 96.171 years apprimately.

b) Plugging y=1200 in given function, we get

1200 = 5.5 (1.025)×

Dividing both sides by 5.5

1200/5.5 = 5.5 (1.025)× / 5.5

1200/5.5 = (1.025)×

Taking natrual log on both sides, we get

ln (1200/5.5) = ln (1.025)×

ln (1200/5.5) = x ln (1.025)

Dividing both sides by ln(1.05), we get

$\frac{ln\frac{1200}{5.5}}{ln(1.05)}=x$

x= 110.377 years apprximately.

Therefore, it will take 96.171 years apprimately to population to reach 600 rabbits and 110.377 years approximately to population to reach 1200 rabbits.

Answer 2

Answer:

Given function that shows the population of the rabbits after x years,

$y=5.5(1.025)^x-----(1)$

Where, 5.5 ( in hundred ) shows the initial population,

When the population is 600,

⇒ y = 6.6,

Then from equation (1),

$6.0=5.5(1.025)^x$

$\frac{6}{5.5}=(1.025)^x$

Taking log on both sides,

$log \frac{6}{5.5} = log (1.025)^x$

$log \frac{6}{5.5} = xlog (1.025)$    $(log m^n = nlog m)$

$\implies x = \frac{log\frac{6}{5.5}}{log 1.025}\implies x=3.5237\approx 4$

Hence, it will take approximately 4 years to reach 600 rabbits.

Now, if the population is 1200,

⇒ y = 12,

Then from equation (1),

$12=5.5(1.025)^x$

$\frac{12}{5.5}=(1.025)^x$

Taking log on both sides,

$log (\frac{12}{55}) = log (1.025)^x$

$log (\frac{12}{55})= xlog (1.025)$

$\implies x = \frac{log (\frac{12}{55})}{log 1.025}\implies x=31.595\approx 32$

Hence, it will take approximately 32 years to reach 1200 rabbits.