Question

"The relationship between the elapsed time t, in years, since Rosario started studying the population, and the number of penguins, P(t), is modeled by the following function:

P(t) = 21,000 / (4t^3).

Complete the following sentence about the rate of change in the penguin population:

The population of penguins loses ____ of its size every ____ years."

Answer 1

The population of penguins loses a fraction of its size every year as described by the given function.

Step-by-step explanation:

The given function is P(t) = 21,000 / (4t^3). The rate of change in the penguin population can be determined by finding the derivative of the function with respect to time, t. The derivative of the function is dP/dt = -21,000 / (4t^4).

This means that the population of penguins loses a fraction of its size every year as t increases. As t increases, the denominator, 4t^4, becomes larger and the derivative becomes more negative. This indicates that the rate of change of the population is negative, and thus the population is decreasing. The statement can be completed as follows: The population of penguins loses an increasing fraction of its size every year. The fraction of the size lost depends on the value of t.