Science:Math Exam Resources/Courses/MATH104/December 2014/Question 03 (a)

MATH104 December 2014

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Question 03 (a)
The price $p$ (in dollars) and the demand $q$ for a product are related by the following demand equation: $p^{3}+q+q^{3}=38$ Find the price elasticity of demand in terms of $p$ and $q$ for this product.

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Hint 1
The price elasticity of a product is given by $\epsilon ={\frac {dq}{dp}}{\frac {p}{q}}$.

Hint 2
Apply implicit differentiation.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall that the price elasticity of a product is given by $\epsilon ={\frac {dq}{dp}}{\frac {p}{q}}$, where $p$ and $q$, are price and demand respectively. We implicitly differentiate the given relation with respect to $p$ giving: ${\begin{aligned}3p^{2}+{\frac {dq}{dp}}+3q^{2}{\frac {dq}{dp}}&=0\\{\frac {dq}{dp}}+3q^{2}{\frac {dq}{dp}}&=-3p^{2}\\{\frac {dq}{dp}}(1+3q^{2})&=-3p^{2}\\{\frac {dq}{dp}}&={\frac {-3p^{2}}{1+3q^{2}}}\end{aligned}}$ Substitute into the formula for elasticity and obtain: $\epsilon ={\frac {-3p^{2}}{1+3q^{2}}}{\frac {p}{q}}={\frac {-3p^{3}}{q+3q^{3}}}$.

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