A colony of 1000 weevils grows exponentially to 1750 in one week. a) At what continuous rate is the population growing?
b) How many weeks does it take for the weevil population to reach 3000?
a) 1750 = 1000 e^(r * 1) ... r is weevils per week
ln(1750 / 1000) = r
b) 3000 = 1000 e^(r * t)
... plug in r ... solve for t (in weeks)
To find the continuous rate at which the weevil population is growing, we can use the formula for exponential growth:
P(t) = P₀ * e^(rt)
Where:
- P₀ is the initial population (1000 weevils in this case)
- P(t) is the population at time t
- r is the continuous growth rate
- e is the base of the natural logarithm
a) To find the continuous growth rate, we can rearrange the formula as follows:
P(t) = P₀ * e^(rt)
1750 = 1000 * e^(r * 7)
Divide both sides of the equation by 1000:
1.75 = e^(7r)
Take the natural logarithm of both sides:
ln(1.75) = 7r
Divide both sides by 7:
r = ln(1.75)/7
Using a calculator, we find that r is approximately 0.0867.
b) Now we can determine how many weeks it takes for the weevil population to reach 3000. We use the same formula:
P(t) = P₀ * e^(rt)
3000 = 1000 * e^(0.0867 * t)
Divide both sides by 1000:
3 = e^(0.0867 * t)
Take the natural logarithm of both sides:
ln(3) = 0.0867 * t
Divide both sides by 0.0867:
t = ln(3)/0.0867
Using a calculator, we find that t is approximately 25.53 weeks.
To calculate the continuous rate of growth, we can use the exponential growth formula, which is:
A = P * e^(rt)
where:
A = final population
P = initial population
r = continuous growth rate
t = time in weeks
e = the base of the natural logarithm (approximately equal to 2.71828)
a) To find the continuous rate of growth, we need to solve the exponential growth formula for r. Let's substitute the given values into the formula:
1750 = 1000 * e^(r * 1)
Dividing both sides by 1000, we get:
1.75 = e^r
To solve for r, we can take the natural logarithm of both sides:
ln(1.75) = r
Using a calculator, we find that ln(1.75) ≈ 0.5596. Therefore, the continuous rate of growth is approximately 0.5596.
b) To find the number of weeks it takes for the weevil population to reach 3000, we can use the same exponential growth formula and solve for t:
3000 = 1000 * e^(0.5596 * t)
Dividing both sides by 1000, we get:
3 = e^(0.5596 * t)
Taking the natural logarithm of both sides:
ln(3) = 0.5596 * t
Dividing both sides by 0.5596, we get:
t = ln(3) / 0.5596
Using a calculator, we find that ln(3) ≈ 1.0986. Therefore, t ≈ 1.0986 / 0.5596 ≈ 1.9622. Since time is typically measured in whole numbers of weeks, the weevil population will reach 3000 in approximately 2 weeks.