Question

A company is selling a certain commodity ‘x’. The demand function for the commodity is linear. The company can sell 2000 units when the price is Rs. 8 per unit and it can sell 3000 units when the price is Rs. 4 per unit. The Marginal revenue at x = 5 is:

• A. Rs. 79.98
• B. Rs. 15.96
• C. Rs. 16.04
• D. Rs. 80.02

Hint:

When working with linear demand functions, the slope represents how price changes with respect to quantity. The marginal revenue is the derivative of the revenue function and indicates the additional revenue generated from selling one more unit. In this case, the marginal revenue at \( x = 5 \) tells us the change in revenue when the quantity increases by 1 unit. Always remember that marginal revenue is crucial for making pricing and production decisions in economics.

Answer 1

The Correct Option is B

The demand function \( p(x) \) is linear. Using the points \( (2000, 8) \) and \( (3000, 4) \), find the slope \( m \) and intercept \( c \):

\[ m = \frac{4 - 8}{3000 - 2000} = \frac{-4}{1000} = -0.004, \quad c = 8 + 2000(0.004) = 16. \]

Thus, \( p(x) = -0.004x + 16 \). Revenue \( R(x) \) is:

\[ R(x) = x \cdot p(x) = x(-0.004x + 16) = -0.004x^2 + 16x. \]

The Marginal Revenue is:

\[ R'(x) = -0.008x + 16. \]

At \( x = 5 \):

\[ R'(5) = -0.008(5) + 16 = -0.04 + 16 = 15.96. \]

Answer 2

The demand function \( p(x) \) is linear. Using the points \( (2000, 8) \) and \( (3000, 4) \), we can find the slope \( m \) and intercept \( c \):

Step 1: Find the slope \( m \):

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \] Substituting the given points \( (2000, 8) \) and \( (3000, 4) \): \[ m = \frac{4 - 8}{3000 - 2000} = \frac{-4}{1000} = -0.004. \]

Step 2: Find the intercept \( c \):

The linear equation is given by \( p(x) = mx + c \). Using the slope \( m = -0.004 \) and the point \( (2000, 8) \), we can find the intercept \( c \): \[ c = 8 + 2000(0.004) = 8 + 8 = 16. \]

Step 3: Write the demand function \( p(x) \):

Now that we have the slope \( m = -0.004 \) and intercept \( c = 16 \), the demand function is: \[ p(x) = -0.004x + 16. \]

Step 4: Revenue function \( R(x) \):

The revenue function is the product of the quantity \( x \) and the price \( p(x) \). Thus: \[ R(x) = x \cdot p(x) = x(-0.004x + 16) = -0.004x^2 + 16x. \]

Step 5: Find the Marginal Revenue \( R'(x) \):

The marginal revenue is the derivative of the revenue function \( R(x) \) with respect to \( x \): \[ R'(x) = \frac{d}{dx} (-0.004x^2 + 16x) = -0.008x + 16. \]

Step 6: Evaluate \( R'(x) \) at \( x = 5 \):

Substituting \( x = 5 \) into the marginal revenue function: \[ R'(5) = -0.008(5) + 16 = -0.04 + 16 = 15.96. \]

Conclusion: Thus, the marginal revenue at \( x = 5 \) is \( 15.96 \).