Predicting Rabbit Population Growth: An Exponential Model Approach

The initial population of rabbits in a laboratory is 20. After 100 days, the population is 80. The question arises: when will the rabbit population reach 200? This article provides a detailed analysis using the exponential growth model to determine the growth rate and predict the future population size.

Constants in the Model

Initial population (P0): 20 Population after 100 days (P100): 80 Exponential growth formula: (P_t P_0 cdot e^{rt}) Where: (P_t): Population at time (t) (r): Growth rate (t): Time in days

Step 1: Determine the Growth Rate

Using the exponential growth model, the population equation at time (t 100) days is:

(P_{100} 20 cdot e^{100r} 80)

Rearrange the equation to isolate (e^{100r}):

(e^{100r} frac{80}{20} 4)

Take the natural logarithm of both sides:

(100r ln 4)

Solve for (r):

(r frac{ln 4}{100})

Step 2: Find the Time When the Population Reaches 200

Now, set up the equation for when the population reaches 200:

(200 20 cdot e^{rt})

Divide both sides by 20:

(10 e^{rt})

Take the natural logarithm:

(ln 10 rt)

Substitute (r):

(ln 10 frac{ln 4}{100} cdot t)

Solve for (t):

(t frac{100 ln 10}{ln 4})

Calculate (ln 10) and (ln 4):

(ln 10 approx 2.3026)

(ln 4 approx 1.3863)

Substitute these values into the equation:

(t frac{100 cdot 2.3026}{1.3863} approx frac{230.26}{1.3863} approx 166.5)

Conclusion

The population will reach 200 rabbits in approximately 166.5 days from the start. Therefore, the population will reach 200 around day 167.

Alternative Method Using Doubling Time

We can also use the doubling time approach to find the time required for the population to reach 200. Given:

(frac{80}{20} 4) After 100 days, the population has grown by 4 times. In another 50 days, the population will grow by 2 times. In 25 days, the population will grow by 80 (4 times 20). In 12.5 days, the population grows by 40 (2 times 20). 50 12.5 62.5 days.

Therefore, in another 63 days, the rabbit population will be 200.

Reasoning with Doubling Time

Considering the growth rate, the population doubles every 50 days. We use the exponential formula again:

(200 20 cdot 2^x)

(10 2^x)

(log_2 10 log_2 2^x)

(x approx 3.32)

One period is 50 days, so 3.32 periods equals 166 days from the beginning or 66 days from the time we reached 80 rabbits.

Final Answer

The population will reach 200 rabbits in approximately 167 days, given the initial conditions and exponential growth pattern.