A company is selling a certain product. The demand function for the product is linear. The company can sell 2000 units when the price is ₹8 per unit and it can sell 3000 units when the price is ₹4 per unit. Determine the total revenue function.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $R(x)=16x-\frac{x^2}{250}$
As the demand function for the product is linear (given), let
$p = a + bx$ ...(1)
where p is the price per unit and x units is the demand of the product, a and b are constants.
Given $x = 2000$ when $p = 8$ and $x = 3000$ when $p = 4$.
Substituting these values in (1), we get
$8 = a + 2000b$ ...(2)
$4 = a + 3000 b$ ...(3)
Subtracting (2) from (3), we get
$1000b=-4⇒b=-\frac{1}{250}$
Substituting this value of b in (2), we get
$8 = a + 2000(-\frac{1}{250})$
$⇒ 8=a-8⇒ a=16$.
Substituting these values of a and b in (1), we get
$p=16-\frac{1}{250}x$.
The demand function is $p=16-\frac{1}{250}x$.
Total revenue by selling x units of the product
$=px=\left(16-\frac{1}{250}x\right)x=16x-\frac{x^2}{250}$
Hence, the revenue function is $R(x) =16x-\frac{x^2}{250}$