Question

The amount of pollution content added in the air in a city to x diesel vehicles is given by P(x)$=0.005{{x}^{3}}+0.02{{x}^{2}}+30x$. Find the marginal increase in pollution content when 3 diesel vehicles are added.

Answer 1

Hint: To find the marginal increase in pollution content, we should know the definition of marginal increase. We know that the increase in a function when we change a parameter is defined as the difference between the final functional value and the initial functional value. The marginal increase which is slightly different in definition from the marginal increase is defined as the ratio of the increase caused in the function due to the parameter to the change in the parameter. In our question, P(x) is the function and x is the parameter. Let us assume that x is very large compared to the 3 new vehicles added. This leads to a conclusion that marginal increase = \[{{\left. \dfrac{dP}{dx} \right|}_{x=3}}\]. To calculate marginal increase, we should differentiate P(x) with respect to x and substitute the value of x = 3 in \[\dfrac{dP}{dx}\]

Complete step-by-step solution:
In the question, we are given the pollution caused by vehicles as a function of the number of vehicles. We are asked to find the marginal increase in pollution when there is an increase in the number of vehicles by 3 in number.
The marginal increase of a function with respect to a parameter is defined as the ratio of the change in function to the change in the parameter. Mathematically, it is written as
Marginal increase $=\dfrac{f\left( {{x}_{2}} \right)-f\left( {{x}_{1}} \right)}{{{x}_{2}}-{{x}_{1}}}$
When the value of increase in the parameter x is small, the marginal increase is approximated as \[{{\left. \dfrac{df}{dx} \right|}_{x={{x}_{1}}}}\]
The given pollution function is $P\left( x \right)=0.005{{x}^{3}}+0.02{{x}^{2}}+30x$
By differentiating the function, we get
$P'\left( x \right)=0.005\times 3{{x}^{2}}+0.02\times 2x+30$
We are asked to find a marginal increase when there is an increase in vehicles by 3.
So, substituting x = 3 in $P'\left( x \right)=0.005\times 3{{x}^{2}}+0.02\times 2x+30$, we get
$\begin{align}
  & P'\left( 3 \right)=0.015{{\left( 3 \right)}^{2}}+0.04\times 3+30 \\
 & P'\left( 3 \right)=0.015\times 9+0.12+30=30.12+0.135=30.255 \\
\end{align}$
$\therefore $The marginal increase in pollution is 30.255.

Note: Some students make a mistake by confusing between marginal increase and increase. They do the calculation of marginal change $P\left( x+3 \right)-P\left( x \right)$ and end up in wrong answer. Their answer will also be in terms of x and not constant. To avoid this mistake, we should know the slight difference between change and marginal change.