The population of a city is p=2500 e^(kt) where t = 0 represents the year 2000. in 1945 the population was 1350. Find the value of k and use this result to predict the population in the year 2010.
if the year 2000 matches t = 0, then
the year 1945 matches t = -55
so we have 1350 = 2500e^(-55k)
.54 = e^(-55k)
using natural logs,
-55k = ln(.54)
k = .011203384
so the equation is p = 2500e^(.011203384t)
in 2010 , t = 10
so
p = 2500E(.011203384*10)
= 2796
To find the value of k, we need to use the given information that in 1945, the population was 1350.
First, let's find the value of t in 1945. Since t = 0 represents the year 2000, we need to calculate the difference between 1945 and 2000.
t = 1945 - 2000 = -55
Now, substitute the values of p (population) and t into the equation:
1350 = 2500e^(k*(-55))
To solve for k, divide both sides of the equation by 2500:
1350/2500 = e^(k*(-55))
0.54 = e^(-55k)
Take the natural logarithm of both sides of the equation:
ln(0.54) = -55k
Now solve for k:
k = ln(0.54) / -55
Using a calculator, we can find that k is approximately -0.03067.
Now we can predict the population in the year 2010 using this value of k.
t = 2010 - 2000 = 10
p = 2500e^(k*t)
p = 2500e^(-0.03067*10)
Using a calculator, we find that the predicted population in the year 2010 is approximately 2195.
To find the value of k, we can use the given information that in 1945 (t = -55 years from t = 0), the population was 1350.
Using the population formula p = 2500e^(kt), we substitute the values p = 1350 and t = -55:
1350 = 2500e^(k*(-55))
Dividing both sides of the equation by 2500 gives us:
0.54 = e^(-55k)
To solve for k, we need to take the natural logarithm (ln) of both sides of the equation:
ln(0.54) = ln(e^(-55k))
Using the property that ln(e^x) = x, we simplify the equation to:
ln(0.54) = -55k
Now we can solve for k by dividing both sides by -55:
k = ln(0.54) / -55
Using a calculator, we find:
k ≈ -0.032
Now that we have the value of k, we can predict the population in the year 2010 (t = 2010 - 2000 = 10 years).
Using the population formula p = 2500e^(kt), we substitute the values p = ?, k = -0.032, and t = 10:
p = 2500e^(-0.032 * 10)
Evaluating the expression:
p ≈ 2500e^(-0.32)
Using a calculator to calculate e^(-0.32):
p ≈ 2500 * 0.7266
p ≈ 1816.5
Therefore, the predicted population in the year 2010 is approximately 1816.5.