Final answer:
To find the number of years it will take for the population of men and women to be equivalent, we set the population growth equations for each group equal to each other. Solving for t, we find that it will take approximately 5 years.
Step-by-step explanation:
To determine how many years it will take for the population of men and women to be equivalent, we need to calculate the population growth of each group over time. The population of men is decreasing by 1% each year while the population of women is increasing by 10% each year. Let's start by finding the growth equation for each group.
The population of men can be represented by the equation:
Population of men = 5000 * (1 - 0.01)^t
Where t represents the number of years.
The population of women can be represented by the equation:
Population of women = 4000 * (1 + 0.1)^t
We want to find the number of years when the populations of men and women are equal. This means that the
population of men is equal to the population of women:
5000 * (1 - 0.01)^t = 4000 * (1 + 0.1)^t
Simplifying the equation, we get:
(0.99)^t = (1.1)^t
Since the exponential terms are equal, we can ignore the bases and focus on the exponents:
t*log(0.99) = t*log(1.1)
Dividing both sides by log(0.99), we get:
t = t * (log(1.1) / log(0.99))
Simplifying further, we have:
t = log(1.1) / log(0.99)
Using a calculator, we can find that the approximate value of t is 4.58 years.
Therefore, it will take approximately 5 years for the population of men and women to be equivalent. The correct answer is b) 5 years.