Question

The total revenue received (in rupees) from the sale of x units of a product is given by$R(x) = 13{x^2} + 26x + 15$.Find the marginal revenue when x = 7

Answer 1

Hint: Marginal revenue is the rate of change of total revenue with respect to the number of units sold. Hence after differentiating put x=7 to get an answer.

Complete step-by-step answer:
 Step 1: Here we are given the total revenue received from the sale of x units
$R(x) = 13{x^2} + 26x + 15$
We know that the marginal revenue is the rate of change of total revenue with respect to the number of units sold.
It is nothing but we need to differentiate R(x) in terms of x
Step 2 :
Marginal revenue = $\dfrac{{dR(x)}}{{dx}}$
$
   \Rightarrow M.R = \dfrac{{d(13{x^2} + 26x + 15)}}{{dx}} \\
    \\
$…………….(1)
Now lets differentiate each term,
We know that differentiation of ${x^n} = n{x^{n - 1}}$
Using this , differentiation of $13{x^2} = 13(2{x^1}) = 26x$
And differentiation of $26x = 26({x^0}) = 26$
And finally differentiation of a constant is zero.
Using all these in equation (1)
$M.R = 26x + 26$
Step 3 :
In the problem we are asked to find the marginal revenue at x = 7
So lets substitute 7 in the place of x in the above equation
$
  M.R = 26(7) + 26 \\
  {\text{ = }}182 + 26 \\
  {\text{ = }}208 \\
$
Therefore the marginal revenue at x = 7 is Rs . 208

Note: Revenue is how much money a business brings in by selling its goods or services at a certain price.
The starting point for any income statement is revenue that will eventually lead to net income after expenses are deducted.
Total revenue is the full amount of total sales of goods and services. It is calculated by multiplying the total amount of goods and services sold by their prices.
Marginal revenue is the increase in revenue from selling one additional unit of a good or service.
Marginal revenue is important because it measures increases in revenue from selling more products and services. Marginal revenue follows the law of diminishing returns, which states that any increases in production will result in smaller increases in output.