The weekly sales of an item is given by \( q=45-4 p^{2} \). Find the price elasticity of the demand function in terms of the price. \( E(p)= \)

To find the price elasticity of the demand function \( E(p) \), we use the formula: \[ E(p) = \frac{dq}{dp} \cdot \frac{p}{q} \] First, calculate \( \frac{dq}{dp} \). Given \( q = 45 - 4p^2 \), differentiating with respect to \( p \) gives: \[ \frac{dq}{dp} = -8p \] Next, substitute \( q \) back into the elasticity formula: \[ E(p) = (-8p) \cdot \frac{p}{45 - 4p^2} \] This simplifies to: \[ E(p) = \frac{-8p^2}{45 - 4p^2} \] So, the price elasticity of demand function is: \[ E(p) = \frac{-8p^2}{45 - 4p^2} \] --- The concept of price elasticity dates back to the works of economists in the 19th century, like Alfred Marshall, who sought to understand how changes in price affect the quantity of goods demanded. This concept helps to analyze consumer behavior, allowing businesses to intelligently set prices based on demand sensitivity. Understanding elasticity has real-world applications, especially in pricing strategies. If a product's demand is elastic (|E(p)| > 1), lowering prices can boost sales significantly, while inelastic demand (|E(p)| < 1) suggests price hikes may not heavily impact sales volume. This insight can guide firms in revenue maximization.