A certain population of rabbits is doubling its population every 2 months. The original population started with only 20 rabbits. A biologist studying the population found that the equation 1000=20(2)^t shows how many months, t, it will take before there are 1000 rabbits in the population.
How many months will it take for the population to reach 1000 rabbits?
Enter your answer in logarithmic form. For example, if your answer is log42(53), enter it like this: log_42(53)
***I have no clue what the answer is, let alone how to solve this. Can I get some help please. Thank you.***
Is the answer log_2(50)?
First of all the equation should be
1000 = 20(2)^(t/2)
divide both sides by 20
50 = 2^(t/2)
take log of both sides and use log rules
(t/2)log2 = log50
t/2 = log50/log2 , which would be log2 50
t/2 = 5.64..
t = 11.29 months
check:
after 2 months there would be 40
after 4 months there would be 80
after 6 months there would be 160
after 8 months there would be 320
after 10 months there would be 640
after 12 months there would be 1280 , my answer of 11.29 months for 1000 rabbits looks very
reasonable.
Thank you!
Sure, let's break down the problem and explain how to solve it step by step.
We are given that the population of rabbits is doubling every 2 months. This means that the new population after each 2-month period is twice the previous population. So, if we denote the population at time t as P(t), we can write the equation as:
P(t) = P(0) * 2^(t/2)
where P(0) represents the initial population, which is 20 rabbits in this case, and t represents the number of months.
We are also given that we want to find the value of t when the population reaches 1000 rabbits. So, we can set up the equation:
1000 = 20 * 2^(t/2)
To solve for t, we need to isolate it on one side of the equation. Let's solve the equation step by step:
Step 1: Divide both sides of the equation by 20.
1000/20 = 2^(t/2)
Simplifying further:
50 = 2^(t/2)
Now we need to isolate t/2. To do that, we need to take the logarithm base 2 of both sides of the equation:
Step 2: Take the logarithm base 2 of both sides of the equation.
log2(50) = log2(2^(t/2))
Since log2(2^(t/2)) can be simplified to t/2, the equation becomes:
log2(50) = t/2
Now we can solve for t by isolating it on one side of the equation:
Step 3: Multiply both sides of the equation by 2.
2 * log2(50) = 2 * (t/2)
2 * log2(50) = t
The final answer is t = 2 * log2(50).
I hope this explanation helps!