The city planners of bakersfield, CA are studying the current and future needs of the city. The sewage system they have in place now can support 300,000 residents. The current population of the city is 180,000, up from 122,000 eight years ago. Assuming that the population will continue to row at this rate, when will the sewer system need to be updated? How many people will the new have to accomodate for the system to last 15 years before needing to be updated again?
Ok, so I know that 180,000 - 122,000 is 58,000 so 58,000/122,000 = 47.5%
increas in eight years. I'm not sure where to go from here.
Should I be using the formula for exponential growth?
Population is usually exponential growth
P(t)=P(o)e^kt
P(o)=122000
find k from
180000=122000 e^kt
log of both sides
ln(180000)=ln(122000)+ kt
then t=8, solve for k
see if you can do it from there. find t when P(t)=300,000
then find p(15+the t you just found).
Yes, you are correct that you can use the formula for exponential growth to estimate the future population of Bakersfield. The formula for exponential growth is:
P(t) = P(0) * (1 + r)^t
Where:
P(t) is the future population at time t
P(0) is the initial population
r is the growth rate
t is the time period
Let's break down the information given in the question:
Initial population (P(0)) = 122,000
Current population = 180,000
Time period (t) = 8 years
To find the growth rate (r), you can use the formula:
r = (P(t)/P(0))^(1/t) - 1
Substituting the given values:
r = (180,000/122,000)^(1/8) - 1
Calculating this, you will find that the growth rate is approximately 0.0559, or 5.59%.
Now that you know the growth rate, you can use the exponential growth formula to project the future population. However, the question asks for when the sewer system needs to be updated, so you need to determine the population at that time.
Let's say the sewer system needs to be updated after 15 years. Now you can calculate P(15):
P(15) = P(0) * (1 + r)^15
To find the population at which the sewer system needs to be updated again, you need to solve for P(15). You can set up an equation:
P(15) = 300,000
And solve it:
300,000 = 122,000 * (1 + 0.0559)^15
By solving this equation, you will find that the population at which the sewer system needs to be updated again is approximately 477,797, rounding up to the nearest whole number.
Therefore, when the population of Bakersfield reaches around 477,797, the sewer system will need to be updated to accommodate the increasing number of residents.