the population growth of a city is modeled by P(t)=240,360e^.012t where t represents the number of years since 2000
a) what is the yearly growth rate as a percent?
b) find the population in 2015
c) how long, to the nearest year, will it take to reach a population of 350,000?
(a) e^0.012 = 1.012, so 1.2% per year
(b) P(15)=240,360e^(.012*15)
(c) just solve 240360 e^.012t = 350000
t = ln(350000/240360)/0.012 = 31.31
a) The yearly growth rate can be found from the equation P(t) = 240,360e^0.012t. The growth rate is given by the exponent, which in this case is 0.012. To find the percentage growth rate, you multiply the exponent by 100:
Yearly growth rate = 0.012 * 100 = 1.2%
Therefore, the yearly growth rate is 1.2%.
b) To find the population in 2015, we need to substitute t = 2015 - 2000 = 15 in the population growth equation:
P(15) = 240,360e^0.012(15)
P(15) = 240,360e^0.18
P(15) ≈ 240,360 * 1.196826
Rounding to the nearest integer, the population in 2015 is approximately:
P(15) ≈ 287,484
c) We need to solve the equation P(t) = 350,000 to find the time it will take to reach a population of 350,000. Substituting P(t) = 350,000 into the population growth equation, we have:
350,000 = 240,360e^0.012t
Dividing both sides by 240,360 gives:
1.453091 ≈ e^0.012t
Taking the natural logarithm (ln) of both sides:
ln(1.453091) ≈ ln(e^0.012t)
Using the property ln(e^x) = x:
0.373950 ≈ 0.012t
Dividing both sides by 0.012 gives:
31.1625 ≈ t
Rounding to the nearest year, it will take approximately 31 years to reach a population of 350,000.
a) To find the yearly growth rate as a percent, we need to look at the exponential equation P(t) = 240,360e^(0.012t). In this equation, the value 0.012 represents the growth rate. To convert it to a percentage, we can multiply it by 100.
So, the yearly growth rate as a percent is 0.012 * 100 = 1.2%.
b) To find the population in 2015, we need to substitute the value of t into the equation. Since t represents the number of years since 2000, we can calculate 2015 - 2000 = 15.
P(t) = 240,360e^(0.012t)
P(15) = 240,360e^(0.012 * 15)
P(15) = 240,360e^(0.18)
Using a calculator, you can find the value of P(15) ≈ 265,985.
Therefore, the population in 2015 is approximately 265,985.
c) To find how long it takes to reach a population of 350,000, we need to set up the equation:
350,000 = 240,360e^(0.012t)
To solve for t, we need to isolate the exponent and solve the equation algebraically. Divide both sides of the equation by 240,360 to get:
350,000/240,360 = e^(0.012t)
Next, take the natural logarithm (ln) of both sides to remove the exponential:
ln(350,000/240,360) = 0.012t
Now divide both sides of the equation by 0.012:
t ≈ ln(350,000/240,360) / 0.012
Using a calculator, you can find t ≈ 9.29.
Therefore, it will take approximately 9 years (rounded to the nearest year) to reach a population of 350,000.