Question

It is estimated that t years from now the tree plantation of a certain forest will be increasing at the rate of 3t ² + 5t + 6 hundred trees per year. Environmentalists have found that the level of Oxygen in the forest increases at the rate of approximately 4 units per 100 trees. By how much will the Oxygen level in the forest increase during the next 3 years?

Answer 1

Assume P(t) is the rate of the tree plantation.

Consider the rate of change of the tree plantation with respect to time t is,

$\frac{dP(t)}{dt} = 3t^2 + 5t +6$ hundred trees per year

The rate function P(t) is an antiderivative of $3t^2 + 5t +6$

During the next 3 years, the rate change is given by the definite integral

$P(3) -P(0) = \int^3_0 \frac{dP(t)}{dt} dt \\ = \int^3_0 3t^2 +5t + 6 dt \\ = [t^3 + 2.5t^2 + 6t]^3_0 \\ = [3^3 + 2.5 \times 3^2 + 6 \times 3 -0] \\ = [27+22.5 + 18] \\ = 67.5 \; hundred \;trees$

Oxygen in the forest increases at the rate of approximately 4 units per 100 trees.

Proportion:

4 units – 100 trees

x units – 6750 trees

$x = \frac{4 \times 6750}{100}=270 \;units$

Therefore, the increased Oxygen level in the forest is 270 units.