Revenue function ‘R’ and cost function ‘C’ are R = 14x – x2

1.	Revenue function ‘R’ and cost function ‘C’ are R = 14x – x2 and C = x(x2 – 2). Find the (i) average cost (ii) marginal cost (iii) average revenue and (iv) marginal revenue.
Answer» R = 14x – x² and C = x(x² – 2) C = x³ – 2x (i) Average Cost (AC) = \(\frac{Total\,cost}{Output}\) = \(\frac{C(x)}{x}\) = \(\frac{x^3-2x}{x}\) = \(\frac{x^3}{x}-\frac{2x}{x}\) = x² - 2 (ii) Marginal Cost (MC) = \(\frac{dC}{dx}\) = \(\frac{d}{dx}\)(x³ – 2x) = \(\frac{d}{dx}\)(x³) – 2 \(\frac{d}{dx}\)(x) = 3x² – 2 (iii) Average Revenue R = 14x – x² Average Revenue (AR) = \(\frac{Total\,Revenue}{Output}=\frac{R(x)}{x}\) = \(\frac{14x-x^2}{x}\) = \(\frac{14x}{x}-\frac{x^2}{x}\) = 14 - x (iv) Marginal Revenue (MR) = \(\frac{dR}{dx}\) = \(\frac{d}{dx}\)(14x – x²) = 14 \(\frac{d}{dx}\)(x) – \(\frac{d}{dx}\)(x²) = 14(1) – 2x = 14 – 2x

Revenue function ‘R’ and cost function ‘C’ are R = 14x – x2 and C = x(x2 – 2). Find the (i) average cost (ii) marginal cost (iii) average revenue and (iv) marginal revenue.

Answer»

R = 14x – x² and C = x(x² – 2)

C = x³ – 2x

(i) Average Cost (AC) = \(\frac{Total\,cost}{Output}\) = \(\frac{C(x)}{x}\)

= \(\frac{x^3-2x}{x}\)

= \(\frac{x^3}{x}-\frac{2x}{x}\)

= x² - 2

(ii) Marginal Cost (MC) = \(\frac{dC}{dx}\)

= \(\frac{d}{dx}\)(x³ – 2x)

= \(\frac{d}{dx}\)(x³) – 2 \(\frac{d}{dx}\)(x)

= 3x² – 2

(iii) Average Revenue R = 14x – x²

Average Revenue (AR) = \(\frac{Total\,Revenue}{Output}=\frac{R(x)}{x}\)

= \(\frac{14x-x^2}{x}\)

= \(\frac{14x}{x}-\frac{x^2}{x}\)

= 14 - x

(iv) Marginal Revenue (MR) = \(\frac{dR}{dx}\)

= \(\frac{d}{dx}\)(14x – x²)

= 14 \(\frac{d}{dx}\)(x) – \(\frac{d}{dx}\)(x²)

= 14(1) – 2x

= 14 – 2x