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Question
The price elasticity of demand for a commodity is `"p"/x^3`. Find the demand function if the quantity of demand is 3 when the price is ₹ 2.
Solution
Price elasticity of demand
`eta_"d" = "p"/x^3`
`(-"p")/x ("d"x)/"dp" = "p"/x^3`
`(-x^3)/x "d"x = "p"/"p" "dp"`
`- x^2 "d"x = "dp"`
Integrating on both sides
`- int x^2 "d"x = int "dp"`
`(-x^3)/3 = "p"+ "k"` ........(1)
When x = 3 and p = 2
`(-(3)^3)/3 = 2 + "k"`
`(- 27)/3 = 2 + "k"`
⇒ – 9 = 2 + k
∴ k = – 11
Equation (1)
⇒ `(-x^3)/3 = "p" - 11`
∴ p = `11 - x^3/3`
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