Question

In 1985 a biologist counted 750 pine trees in a 250 hectare forest. Using similar counting techniques, the biologist counted 1,250 pine trees in 1990 and 1,500 pines in 1995.
What was the average change of the size of the population from 1985 to 1995?

What was the density of pine trees each year that they were counted?

What was the average change of density from 1985 to 1995?

Answer 1

Answer:

Suppose that we have a given function f(x)

The average rate of change of the function between two values x₁ and x₂ is given by:

$r = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

a) We want to find the average (rate) of change on the size of population from 1985 to 1995.

We have that:

f(1985) = 750

f(1995) = 1500

Then we have:

$r = \frac{1500 - 750}{1995 - 1985} = 750/10 = 75$

This means that the population of trees increases, in average, at a rate of 75 trees per year.

b) What is the density of trees each year that they were counted?

This will be equal to the quotient between the number of trees and the area.

1985: number of trees = 750 pines

         area = 250 ha

Then the density is:

D(1985) = (750 pines)/(250 ha) = 3 pines/ha

So 1985, there were 3 pines per hectare.

1990: number of trees = 1250 pines

         area = 250 ha

Then the density is:

D(1990) = (1250 pines)/(250 ha) = 5 pines/ha

1995: number of trees = 1500 pines

         area = 250 ha

The density is:

D(1995) = (1500 pines)/(250 ha) = 6 pines/ha

3) now we want to get the average change between 1985 and 1995 in the density, this will be:

$r = \frac{D(1995) - D(1885)}{1995 - 1985} = \frac{6 pines/ha - 3pines/ha}{10} = 0.3 pines/ha$

So, on average, each year the number of pines per hectare increases by 0.3