With over 30 million rabbits, the bunny farm is getting overcrowded and some of the rabbits are dying from a contagious disease. The rabbits have stopped reproducing, and the disease is reducing the total rabbit population at a rate of about 30% each month. If this continues, then in how many months will the population drop below 100 rabbits? (For t(0), use your number from part b.)
30,000,000 * 0.7^t < 100
0.7^t < 1/300,000
t > -log(300,000)/log(0.7)
t > 35.36
100%-30%= 70%, so 70/100= 0.7
0.7 is your multiplier that mean he started at 30,000,000 rabbits which is your initial value. So the equation looks like this. t(n)= 30,000,000*(0.7)^n
t(n)= 30,000,000*(0.7)^36
t(n)= 79.5519253757896 which is estimated by 79.56
Therefore, in the month of 36 is where the population of rabbits will be below 100 rabbits.
To find out how many months it will take for the population to drop below 100 rabbits, we can use the exponential decay formula:
N(t) = N0 * e^(rt)
Where:
N(t) is the population at time t
N0 is the initial population
r is the exponential decay rate
t is the time in months
We're given that the initial population (N0) is 30 million rabbits and the decay rate (r) is 30% per month. Therefore, we can rewrite the formula as:
N(t) = 30,000,000 * e^(0.3t)
We need to find the value of t when N(t) drops below 100 rabbits.
100 = 30,000,000 * e^(0.3t)
Dividing both sides by 30,000,000:
0.000003333 = e^(0.3t)
Taking the natural logarithm (ln) of both sides:
ln(0.000003333) = 0.3t
Using a calculator, we can find the value of ln(0.000003333) to be approximately -12.60. Therefore:
-12.60 = 0.3t
Dividing both sides by 0.3:
t = -12.60 / 0.3
t ≈ -42
Since time cannot be negative, we can conclude that it will take approximately 42 months for the population to drop below 100 rabbits.
To determine in how many months the rabbit population will drop below 100, we need to set up an equation based on the given information.
Let's denote the current rabbit population as P(t), where t represents the number of months that have passed.
We know that the population is reducing at a rate of 30% per month, so each month the population will be 70% (or 0.7) of the previous month's population:
P(t) = 0.7 * P(t-1)
We also know that initially there were 30 million rabbits, so we can write:
P(0) = 30,000,000
We want to find the number of months it takes for the population to drop below 100, so we set up the equation:
P(t) < 100
Now, let's solve this equation to find the number of months it takes for the population to drop below 100 rabbits.
Plug in the given initial population P(0) = 30,000,000 into the equation:
P(0) = 0.7 * P(-1)
30,000,000 = 0.7 * P(-1)
Now, let's solve for P(-1):
P(-1) = 30,000,000 / 0.7
P(-1) ≈ 42,857,143
We can see that the population drops below 100 at P(-1), which is less than 100. Therefore, the answer is t = 0 months.
Hence, the population will drop below 100 rabbits immediately (at t = 0).