Question

According to the Florida Agency for Workforce, the monthly average number of unemployment claims in a certain county is given by ????????(tt) = 24.31tt2 − 276.58tt + 2035, where t is the number of years after 1990. a) During what years did the number of claims decrease? b) Find the relative extrema and interpret it.

Answer 1

a) The number of claims decrease from 1990 to 1996.

b) The relative extrema is a minimum and happens approximately in 1996 (t=5.688). This means the moment when the number of claims stop decreasing and start to increase.

Explanation:

The monthly average number of unemployment claims in a certain county is given by:

`C(t)=24.31t^2-276.58t+2035`

With t: number of years after 1990.

We have to determine in what years the number of claims decrease and the relative extreme value.

We can find this by analizing the first derivative.

When the first derivative is equal to zero, this indicates an extreme value, which can be a maximum or minimum.

When the first derivative is positive, it indicates that the function is increasing. On the contrary, when the first derivative is negative, it indicates that the function is decreasing.

The first derivative is:

`(dC)/(dt)=24.31(2t)-276.58(1)+0\\\\\\(dC)/(dt)=48.62t-276.58`

Then, we can calculate the extreme value:

`(dC)/(dt)=48.62t-276.58=0\\\\\\48.62t=276.58\\\\\\t=(276.58)/(48.62)=5.688\approx 6`

This extreme value happens for t=6 (year 1996).

If we calculate the value of the first derivative for t=5, that is previous to the extreme value, we can find if the function was increasing or decreasing:

`(dC)/(dt)(5)=48.62*5-276.58=243.10-276.58=-33.48<0`

As the value is negative, we know that the number of claims was decreasing from t=0 to t=6 (from 1990 to 1996), and then reach a minimum and start to increase from them (from 1996 onwards).