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Differential Equations

Question
CBSEENMA12035733

The amount of pollution content added in air in city due to x-diesel vehicles is given by straight P left parenthesis straight x right parenthesis space equals space 0.005 straight x cubed plus 0.02 straight x squared plus 30 straight x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above questions.

Solution
straight P left parenthesis straight x right parenthesis space equals space 0.005 straight x cubed plus 0.02 straight x squared plus 30 straight x
Differentiating w.r.t.x,
Marginal increase in pollution content equals fraction numerator dP left parenthesis straight x right parenthesis over denominator dx end fraction equals 0.015 straight x squared plus 0.04 straight x plus 30
Putting x = 3 in (1), open parentheses fraction numerator dp left parenthesis straight x right parenthesis over denominator dx end fraction close parentheses subscript straight x equals 3 end subscript space equals space 0.015 cross times 9 plus 0.04 cross times 3 plus 30 equals 30.255
Therefore, the value of marginal increase pollution content is 30.255.
Increase in number of vehicles increases the pollution. We should aim at saving the environment by reducing the pollution by decreasing the vehicle density on road. 

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