The average variable cost of the monthly output of x tonnes of a firm

1.	The average variable cost of the monthly output of x tonnes of a firm producing a valuable metal is Rs \(\frac{1}{5}\)x2 – 6x + 100. Show that the average variable cost curve is a parabola. Also, find the output and the average cost at the vertex of the parabola.
Answer» Let output be x and average variable cost = y y = \(\frac{1}{5}\)x² – 6x + 100 ⇒ 5y = x² – 30x + 500 ⇒ x² – 30x + 225 = 5y – 500 + 225 ⇒ (x – 15)² = 5y – 275 ⇒ (x – 15)² = 5(y – 55) which is of the form X² = 4(\(\frac{5}{4}\))Y ∴ Y average variable cost curve is a parabola Vertex (0, 0) x – 15 = 0; y – 55 = 0 x = 15; y = 55 At the vertex, output is 15 tonnes and average cost is Rs 55.

The average variable cost of the monthly output of x tonnes of a firm producing a valuable metal is Rs \(\frac{1}{5}\)x2 – 6x + 100. Show that the average variable cost curve is a parabola. Also, find the output and the average cost at the vertex of the parabola.

Answer»

Let output be x and average variable cost = y

y = \(\frac{1}{5}\)x² – 6x + 100

⇒ 5y = x² – 30x + 500

⇒ x² – 30x + 225 = 5y – 500 + 225

⇒ (x – 15)² = 5y – 275

⇒ (x – 15)² = 5(y – 55) which is of the form X² = 4(\(\frac{5}{4}\))Y

∴ Y average variable cost curve is a parabola Vertex (0, 0)

x – 15 = 0; y – 55 = 0

x = 15; y = 55

At the vertex, output is 15 tonnes and average cost is Rs 55.