Suppose you know that a population,P, of termites is growing continuously in an exponential manner. (Use the formant P=P0e^kt) If the pest control service estimates that the number of termites will double in 6 months...
A) Determine the monthly continuous population growth rate k as a decimal to 4 places.
B) If the current number of termites is 4000, determine the population in 3 months. Use the rounded value k. Round your answer to the nearest whole termite.
C) The pest control service states that their treatment method will cut the number of termites in half every 10 days. Using the function type P=P0e^kt, determine the daily continuous decay rate,k,for the termite population under this treatment. Round to 4 places.
ln(2) = 0.69314718...
(A) P(6)/P(0) = 2
.: k_g = ln(2) /6mth
(B) P(3mt) = 4000 e^{3 k_g}
(C) k_d = -ln(2) /10day
A) To determine the monthly continuous population growth rate k, we can use the given information that the number of termites will double in 6 months.
In the formula P = P₀e^(kt), where P is the population at time t, P₀ is the initial population, k is the growth rate constant, and t is the time in months, we can substitute the given values:
P = P₀e^(kt)
P/P₀ = e^(6k) (since the population doubles, P = 2P₀, so P/P₀ = 2)
2 = e^(6k)
To solve for k, we can take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(6k))
ln(2) = 6k ln(e)
ln(2) = 6k
Now, we can solve for k by dividing both sides of the equation by 6:
k = ln(2) / 6
Using a calculator (rounded to 4 decimal places), the monthly continuous population growth rate k is approximately 0.1155.
B) To determine the population in 3 months, we can use the formula P = P₀e^(kt) again.
Given that the current number of termites is 4000 (P₀ = 4000), and we have already calculated the growth rate k to be 0.1155, we can substitute these values into the formula:
P = 4000e^(0.1155 * 3)
Using a calculator, we find that the population in 3 months is approximately 5325 termites.
C) The treatment method is stated to cut the number of termites in half every 10 days. This corresponds to a decay rate, since the population is decreasing.
Using the formula P = P₀e^(kt), where P is the population at time t, P₀ is the initial population, k is the decay rate constant, and t is the time in days, we need to determine the daily continuous decay rate k.
If the number of termites is cut in half every 10 days, it means that P/P₀ = 1/2.
Substituting these values into the formula, we have:
1/2 = e^(k * 10)
To solve for k, we take the natural logarithm of both sides:
ln(1/2) = ln(e^(k * 10))
ln(1/2) = 10k ln(e)
ln(1/2) = 10k
Now, divide both sides by 10 to find k:
k = ln(1/2) / 10
Using a calculator (rounded to 4 decimal places), the daily continuous decay rate k is approximately -0.0693. Note the negative sign, indicating decay instead of growth.