The population of a colony of rabbits grows exponentially. The colony begins with 5 rabbits; 5 years later there are 310 rabbits.
(a) Express the population of the colony of rabbits, P , as a function of time, t , in years.
(b) Use the graph to estimate how long it takes for the population of rabbits to reach 1000 rabbits.
(a) Well, I guess you could say that rabbits really know how to multiply... exponentially! To express the population of the colony of rabbits as a function of time, we can use the general equation for exponential growth: P = P₀ * e^(kt). Since the colony begins with 5 rabbits, we can plug that in as our initial population, and we know that after 5 years there are 310 rabbits, so we can plug that in as well.
So, we have P₀ = 5 and P = 310. Let's solve for k!
310 = 5 * e^(k * 5)
Divide both sides by 5:
62 = e^(5k)
Now take the natural logarithm of both sides:
ln(62) = 5k
Finally, divide both sides by 5:
k = ln(62)/5
Now we have k, we can plug it back into the equation to find the population as a function of time!
P = 5 * e^[(ln(62)/5) * t]
(b) Now, estimating how long it takes for the population of rabbits to reach 1000 rabbits, let's plug in P = 1000 and solve for t:
1000 = 5 * e^[(ln(62)/5) * t]
Divide both sides by 5:
200 = e^[(ln(62)/5) * t]
Take the natural logarithm of both sides:
ln(200) = (ln(62)/5) * t
Now, divide both sides by ln(62)/5:
t = ln(200) / (ln(62) / 5)
Plug this equation into a calculator, and you'll get your estimated time! Just remember, rabbits can be unpredictable, so this is just an estimate. Don't be hopping mad if it's not exact!
To express the population of the colony of rabbits, P, as a function of time, t, we can use the formula for exponential growth:
P(t) = P0 * e^(rt)
where P0 is the initial population, r is the growth rate, and e is the base of the natural logarithm.
(a) We know that the colony begins with 5 rabbits, so P0 = 5. We also know that 5 years later there are 310 rabbits. So, we can use this information to find the value of r.
P(5) = 310
5 = 5 * e^(5r)
Divide both sides by 5 to isolate the exponential term:
1 = e^(5r)
Take the natural logarithm (ln) of both sides to solve for r:
ln(1) = ln(e^(5r))
0 = 5r
Divide both sides by 5 to find r:
0 = r
So, the growth rate, r, is 0.
Now that we have the growth rate, we can express the population of the colony of rabbits as a function of time:
P(t) = 5 * e^(0 * t)
P(t) = 5
The population will remain constant at 5 rabbits over time.
(b) Since the population remains constant at 5 rabbits, it will never reach 1000. Therefore, it would take an infinite amount of time for the population of rabbits to reach 1000.
To find the function that expresses the population of the colony of rabbits as a function of time, we can use the exponential growth formula:
P(t) = P0 * e^(kt)
Where:
P(t) = population at time t
P0 = initial population
k = growth rate
e = base of the natural logarithm (approximately 2.718)
In this case, the initial population is 5 rabbits and it grows to 310 rabbits after 5 years.
We can use this information to solve for k in the exponential growth formula.
310 = 5 * e^(5k)
To solve for k, we can divide both sides by 5 and take the natural logarithm of both sides:
62 = e^(5k)
ln(62) = 5k
Now we can solve for k,
k = ln(62) / 5 ≈ 0.970
Now we can write the function to express the population of the colony of rabbits as a function of time:
P(t) = 5 * e^(0.970t)
To estimate how long it takes for the population of rabbits to reach 1000, we can use the equation P(t) = 1000 and solve for t:
1000 = 5 * e^(0.970t)
Dividing both sides by 5:
200 = e^(0.970t)
Taking the natural logarithm of both sides:
ln(200) = 0.970t
Now we can solve for t,
t = ln(200) / 0.970 ≈ 5.44
Therefore, it takes approximately 5.44 years for the population of rabbits to reach 1000.
P = A e^kt
A = 5 is given when t = 0 (because e^0 = 1)
P = 5 e^kt
310 = 5 e^5k
62 = e^5k
ln 62 = 5 k
5 k = 4.127
k = .8254
so
P = 5 e^.8254 t
now if P = 1,000
200 = e^.8254 t
ln 200 = 5.298 = .8254 t
t = 6.42 years