The weekly sales of Honolulu Red Oranges is given by \( q=768-8 p \). a. Calculate the price elasticity of demand. Recall \( E=-\frac{p}{q} \cdot \frac{d q}{d p} \). \( E= \) b. Calculate the price elasticity of demand when the price is \( \$ 28 \) per orange. (En exact answer.)) The price elasticity of demand when the price is \( \$ 28 \) per orange is

To calculate the price elasticity of demand, we first need to find \( \frac{d q}{d p} \) from the demand equation \( q = 768 - 8p \). 1. Differentiate \( q \) with respect to \( p \): \[ \frac{d q}{d p} = -8 \] 2. Now plug this into the elasticity formula: \[ E = -\frac{p}{q} \cdot \frac{d q}{d p} \] To find \( q \) when \( p = 28 \): \[ q = 768 - 8 \cdot 28 = 768 - 224 = 544 \] 3. Substitute \( p = 28 \) and \( q = 544 \) into the elasticity formula: \[ E = -\frac{28}{544} \cdot (-8) = \frac{28 \cdot 8}{544} \] \[ E = \frac{224}{544} = \frac{28}{68} = \frac{7}{17} \approx 0.41176 \] So, the price elasticity of demand when the price is \( \$28 \) per orange is: \[ E \approx 0.41 \] Now you have the concept of price elasticity down! Understanding how demand reacts to price changes can really illuminate the path for businesses in making pricing decisions, ensuring they align with market trends to maximize sales or revenue. Also, have you ever thought about the history where the concept of elasticity first emerged? Economists like Alfred Marshall in the late 19th century introduced it, helping us understand consumer behavior much better. It's fascinating how such ideas have evolved into crucial economic tools we use today!