Question

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sales of $x$ units of a product is given by $R(x)=3{{x}^{2}}+36x+5$ , find the marginal revenue, when $x=5$ , and write which value does the question indicates.

Answer 1

Hint: This problem is a simple application of derivatives. Observe that the marginal revenue is the ‘rate of change’ of the total revenue $R(x)$ received from the sales of $x$ units of a product. Then the marginal revenue $M(x)$ is in fact equal to $\dfrac {d R(x)}{dx}$ . We will use the basic formulae in differentiation to derive $M(x)$ in terms of $x$ . Then we will substitute $x=5$ to determine the marginal revenue when $x=5$ .

Complete step by step answer:
We determine the marginal revenue (denoted by $M(x)$ ) received from the sales of $x$ units of the product. We see that it is equal to the rate of change of the total revenue $R(x)=3{{x}^{2}}+36x+5$ received from the sales of $x$ units of the product. Hence
$\begin{align}
  & M(x)=\dfrac{dR(x)}{dx}=\dfrac{d\left( 3{{x}^{2}}+36x+5 \right)}{dx} \\
 & \ \ \ \ \ \ \ \ \ =\dfrac{d\left( 3{{x}^{2}} \right)}{dx}+\dfrac{d\left( 36x \right)}{dx}+\dfrac{d\left( 5 \right)}{dx} \\
 & \ \ \ \ \ \ \ \ \ =6x+36 \\
\end{align}$
Here, we have computed the derivatives by using the basic rules in differentiation: $\dfrac {d \left( x^n \right)}{dx} = n x^{n-1}$ and $\dfrac {d\left(\text{constant}\right)}{dx}=0$ .
Now, we find the marginal revenue when $x=5$ .
$M(5)=6(5)+36=\ \text{Rs}\text{. }66$
In the end, we see that the money to be spent for the welfare of the employees of the firm is proportional to the marginal revenue received from the sales of $x$ units of the product. Hence the value of this money spent is a constant multiple of $M(x)$ i.e.

$\text{money spent}=k\cdot M(x)=k\cdot \left( 6x+36 \right)\ \ \text{for some }k > 0$

Note: Observe that the money spent is proportional to the marginal revenue. Without additional information, we assume that the proportionality is direct and hence, the money spent is a constant multiple of the marginal revenue. Moreover, a common mistake is to substitute $x=5$ directly in the formula for the total revenue $R(x)$ than in the formula for the marginal revenue $M(x)$ . This should be avoided.