Question

Following the lead of the National Wildlife Federation, the Department of the Interior of a South American country began to record an index of environmental quality that measured progress and decline in the environmental quality of its forests. The index for the years 2004 through 2014 is approximated by the functionI I(t) = 1/3t^3 − 5/2t^2 + 80 (0 ≤ t ≤ 10) where t = 0 corresponds to 2004. For what years is the quality of the forests in this country improving?

Answer 1

After 2009 the quality of the forests in this country improving.

Explanation:

Consider the provided function.

`f(t) = (1)/(3)t^3-(5)/(2)t^2 + 80`

`f'(t) = t^2-5t`

Substitute f'(t)=0.

`t^2-5t=0`

`t(t-5)=0`

`t=0\ or\ t=5`

It is given that (0 ≤ t ≤ 10)

The test interval are (0,5) and (5,10)

For (0 < t < 5) the value of
`f'(t) <0`

For (5 < t < 10) the value of
`f'(t) >0`

Hence, the function is increasing from (5,10).

t = 0 corresponds to 2004.

Therefore, after 2009 the quality of the forests in this country improving.