# In which year was the rainfall maximum and when was it minimum:The rainfall in a city follows the equation R = 3*t^3-12*t^2+3*t where t is the year with t = 0 in 1900.

justaguide | College Teacher | (Level 2) Distinguished Educator

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The rainfall in the city is given by R = 3*t^3-12*t^2+3*t where t is the year with t = 0 in 1900.

To find the years when the rainfall was maximum and minimum, solve `(dR)/(dt)` = 0

`(dR)/(dt)` = 3*t^3-12*t^2+3*t

9t^2 - 24t + 3 = 0

t1 = `(24 + sqrt(24^2 - 108))/18`

=> t1 = `4/3 + sqrt 468/18`

=> t1 = `4/3 + sqrt 13/3`

t2 = `4/3 - sqrt 13/3`

The second derivative of R is R'' = 18t - 24

For t1 = `4/3 + sqrt 13/3` , R'' = `18(4/3 + sqrt 13/3) - 24 ~~ 21.63` which is positive, this indicates that R has a minimum value at t = `4/3 + sqrt 13/3`

for t2 = `4/3 - sqrt 13/3` , R'' = `18(4/3 - sqrt 13/3) - 24 ~~ -21.63` which is negative indicating that R has a maximum at t = `4/3 - sqrt 13/3` .

`4/3 + sqrt 13/3 = 2.53` and `4/3 - sqrt 113/3 = 0.13`

As t = 0 in 1900, and the values of t are in decimals only an approximate year in which rainfall is maximum and minimum can be given.

The rainfall is minimum in 1902 and maximum in 1900