The population of a city has been decreasing exponentially since 1990. In 1990, the population was 1,000,000. In 2010, the population was 560,000. If t represents time in years since 1990, which of the following equations best models the decay of the population?
100000*(56/100)^(t/(2010-1990))
massaged as needed
To find the equation that best models the decay of the population, we need to consider that exponential decay can be modeled using the formula:
P(t) = Pā * e^(-kt)
Where:
P(t) is the population at time t
Pā is the initial population
e is the base of the natural logarithm (approximately 2.71828)
k is the decay constant
We are given that the population in 1990 (at t = 0) is 1,000,000 (Pā), and in 2010 (at t = 20 years), the population is 560,000 (P(20)).
Plugging these values into the equation, we have:
560,000 = 1,000,000 * e^(-k*20)
Now, we can solve for k, which represents the decay constant.
Dividing both sides of the equation by 1,000,000:
0.56 = e^(-k*20)
Taking the natural logarithm of both sides:
ln(0.56) = ln(e^(-k*20))
Using the property of logarithms (ln(a^b) = b * ln(a)):
ln(0.56) = -20k * ln(e)
Since ln(e) is equal to 1, we can simplify further:
ln(0.56) = -20k
Now, divide both sides by -20:
k = ln(0.56) / -20 ā -0.02969
Therefore, the equation that best models the decay of the population is:
P(t) = 1,000,000 * e^(-0.02969t)
This is the exponential decay equation that describes the population's decrease over time since 1990.