A deer population was measured to be 4000. Two years later , it was measured to be 4300. assume the population grows exponentially.
A). Write an equation describe this situation. Find values of all constants.
b) What will the population be four years after the first measurement?
C).How many years will it take the population to double?
present=original*e^kt
4300/4000=e^k*2
ln of both sides..
2k=.073
k=.0365
a) present=4000e^(.0365t)
b. P(6)=4300e^(4*.0365)=4976
c. 2=e^((.0365t)
ln2=.0365 t solve for t. I get about 19 years
A) To describe this situation, we can use the exponential growth model equation:
P(t) = P₀ * e^(r*t)
Where:
- P(t) represents the population at time t
- P₀ represents the initial population
- e is Euler's number (approximately 2.71828)
- r is the growth rate
- t is the time elapsed
In our case, the initial population P₀ is 4000. We need to find the growth rate (r). To do this, we can use the formula for continuous compound interest:
P(t) = P₀ * e^(r*t)
4300 = 4000 * e^(r*2)
Dividing both sides by 4000:
1.075 = e^(2r)
To solve for r, we can take the natural logarithm (ln) of both sides:
ln(1.075) = ln(e^(2r))
ln(1.075) = 2r
Dividing by 2:
r ≈ ln(1.075) / 2 ≈ 0.0368
Therefore, the value of the growth rate (r) is approximately 0.0368.
B) To find the population four years after the first measurement, we can substitute the values into the exponential growth equation:
P(t) = P₀ * e^(r*t)
P(4) = 4000 * e^(0.0368 * 4)
P(4) ≈ 4000 * e^0.1472 ≈ 4000 * 1.158
P(4) ≈ 4632
Therefore, the population will be approximately 4632 four years after the first measurement.
C) To determine how many years it will take for the population to double, we can set up the exponential growth equation and solve for t when P(t) becomes double the initial population:
P(t) = P₀ * e^(r*t)
2 * P₀ = P₀ * e^(r*t)
Dividing both sides by P₀:
2 = e^(r*t)
To solve for t, we can take the natural logarithm of both sides:
ln(2) = ln(e^(r*t))
ln(2) = r*t
Dividing by r:
t ≈ ln(2) / r ≈ ln(2) / 0.0368 ≈ 18.9
Therefore, it will take approximately 18.9 years for the population to double.