the current population of a city is 45000 and is increasing according to P=45000(1+t)^n and n=number of years. if the rate of increase is 7.5% how long will it take the population to double?
90000 = 45000(1.075)^n
2 = 1.075^n
log both sides
log 2 = nlog1.075
n = log2/log1.075 = appr 8.58 years
To find out how long it will take for the population to double, we need to solve the equation P = 2 * 45000 (since we want to find the time it takes for the population to reach double its initial value).
Given that the rate of increase is 7.5%, we can rewrite the equation as:
2 * 45000 = 45000(1 + (7.5/100))^n
Simplifying:
90000 = 45000(1 + 0.075)^n
Dividing both sides by 45000:
2 = (1.075)^n
Now, to solve for n, we can take the logarithm (base 10) of both sides:
log(2) = log((1.075)^n)
Using the logarithmic property, we can bring the power of n down:
log(2) = n * log(1.075)
Finally, we can solve for n by dividing both sides by log(1.075):
n = log(2) / log(1.075)
Using a calculator, we find:
n ≈ 9.781
Therefore, it will take approximately 9.781 years for the population to double.
To find out how long it will take for the population to double, we can use the given equation that relates the population to time:
P = 45000(1 + t)^n
Where:
P is the final population (double the current population)
t is the rate of increase (7.5% = 0.075)
n is the number of years
Since the population needs to double, P will be equal to 2 times the current population:
2 * 45000 = 45000(1 + 0.075)^n
Now we can solve for n:
2 = (1.075)^n
To isolate the exponent, we can take the natural logarithm (ln) of both sides:
ln(2) = ln((1.075)^n)
Using the property of logarithms, we can bring down the exponent:
ln(2) = n * ln(1.075)
Now, to find the value of n, divide both sides of the equation by ln(1.075):
n = ln(2) / ln(1.075)
Using a calculator, we can find that ln(2) ≈ 0.6931 and ln(1.075) ≈ 0.0712. So:
n ≈ 0.6931 / 0.0712
n ≈ 9.727
Therefore, it will take approximately 9.727 years for the population to double.