A population grows from 11,000 to 15,000 in three years.
Enter your answers to three decimal places.
Assuming the growth is exponential, find the growth rate and continuous growth rate
Growth rate ______ %?
Continuous rate _________ %?
I guess by growth rate you mean with yearly compounding and continuous means e^kt
if yearly for three years
15 = 11 (1+r)^3
1.3636 = (1+r)^3
log 1.3636 = 3 log (1+r)
1+r = 1.1089
so
r = .109 = 10.9 percent
now if continuous
1.3636 = e^3k
ln 1.3636 = 3 k
k = .103 or 10.3 percent continuous compounding
Oh wow, the population must have been doing some serious yoga stretches to grow that much! Let's calculate those rates for you:
To find the growth rate, we need to use the formula:
Growth rate = ((Final population - Initial population) / Initial population) * 100
Growth rate = ((15,000 - 11,000) / 11,000) * 100
Growth rate = (4,000 / 11,000) * 100
Growth rate = 0.3636 * 100
Growth rate = 36.36 %
So the growth rate is approximately 36.36 %.
Now moving on to the continuous growth rate. This one requires a different formula:
Continuous growth rate = ln(Final population / Initial population) / number of years
Continuous growth rate = ln(15,000 / 11,000) / 3
Continuous growth rate = 0.3293 / 3
Continuous growth rate = 0.1098
And the continuous growth rate (to three decimal places) is approximately 0.110.
Hope this helps, even if I may have grown some popcorn in the process!
To find the growth rate and continuous growth rate, we can use the formula for exponential growth:
Growth rate = ((P2 / P1)^(1/n) - 1) * 100
Continuous growth rate = ln(P2 / P1) * 100
Where:
P1 = initial population
P2 = final population
n = number of years
Given:
P1 = 11,000
P2 = 15,000
n = 3
Let's calculate the growth rate first:
Growth rate = ((15,000 / 11,000)^(1/3) - 1) * 100
= (1.3636 - 1) * 100
= 0.3636 * 100
= 36.36%
Now, let's calculate the continuous growth rate:
Continuous growth rate = ln(15,000 / 11,000) * 100
= ln(1.3636) * 100
= 0.312 * 100
= 31.2%
Therefore, the growth rate is 36.36% and the continuous growth rate is 31.2%.
To find the growth rate and continuous growth rate, we can use the formula for exponential growth:
\[P(t) = P_0 \times e^{rt}\]
Where:
- \(P(t)\) is the population at time \(t\)
- \(P_0\) is the initial population
- \(r\) is the growth rate
- \(e\) is a mathematical constant approximately equal to 2.71828
Given that the initial population (\(P_0\)) is 11,000 and the population after three years (\(P(3)\)) is 15,000, we can form two equations:
For the initial population:
\[11,000 = P_0 \times e^{r \times 0}\]
Simplifying, we get \(11,000 = P_0\)
For the population after three years:
\[15,000 = 11,000 \times e^{r \times 3}\]
To find the growth rate (\(r\)), we need to solve the second equation for \(r\).
Divide both sides of the equation by 11,000:
\[\frac{15,000}{11,000} = e^{r \times 3}\]
Take the natural logarithm of both sides of the equation:
\[\ln\left(\frac{15,000}{11,000}\right) = r \times 3\]
Simplify:
\[r = \frac{\ln\left(\frac{15,000}{11,000}\right)}{3}\]
Using a calculator, we can find the value of \(r\) and then calculate the growth rate and continuous growth rate.