A population of primates in a certain jungle is doubling each year. Describe how the population increases in size as a function of time. Give some example values for this population over time. Is this a linear or exponential relationship?
To describe how the population increases over time, we need to understand the concept of exponential growth. Exponential growth occurs when a quantity increases by a fixed percentage over a constant time interval. In the case of the primate population doubling each year, we can express the increase as a function of time.
Let's assume the initial population is P₀. Each year, the population doubles, so at the end of the first year, the population would be 2P₀. At the end of the second year, it would be 2(2P₀) = 2²P₀ = 4P₀. Similarly, at the end of the third year, it would be 2(4P₀) = 2³P₀ = 8P₀. This pattern continues, and at the end of the n-th year, the population would be 2ⁿP₀.
Expressing it more generally, the population as a function of time (t) would be given by the equation: P(t) = 2ᵗP₀.
Here are some example values for the population over time, assuming an initial population of 100:
- After 1 year: P(1) = 2¹ * 100 = 200
- After 2 years: P(2) = 2² * 100 = 400
- After 3 years: P(3) = 2³ * 100 = 800
- After 4 years: P(4) = 2⁴ * 100 = 1600
As we can see, the population increases exponentially, doubling each year.
Therefore, the relationship between time and population size in this case is exponential, not linear.