The task presented involves calculating the price elasticity of demand using the given demand function:

$f(p) = 480 - (p + 11)^2$

Steps to Solve

a) Elasticity Formula

The elasticity of demand, denoted by $\varepsilon(p)$, is given by the formula: $\varepsilon(p) = \frac{-f'(p) \cdot p}{f(p)}$

1. Differentiate $f(p)$ with respect to $p$: $f'(p) = \frac{d}{dp}[480 - (p + 11)^2] = -2(p + 11)$

2. Substitute $f'(p)$ into the elasticity formula: $\varepsilon(p) = \frac{-(-2(p + 11)) \cdot p}{480 - (p + 11)^2} = \frac{2(p + 11)p}{480 - (p + 11)^2}$

Thus, the elasticity function is: $\varepsilon(p) = \frac{2(p + 11)p}{480 - (p + 11)^2}.$

b) Elasticity at $p = 3$

1. Substitute $p = 3$ into the formula: $\varepsilon(3) = \frac{2(3 + 11)(3)}{480 - (3 + 11)^2}.$

2. Simplify the terms:

   • $3 + 11 = 14$,
   • $(3 + 11)^2 = 14^2 = 196$,
   • $480 - 196 = 284$.

   So: $\varepsilon(3) = \frac{2(14)(3)}{284} = \frac{84}{284}.$

3. Simplify $\frac{84}{284}$: $\varepsilon(3) \approx 0.296 (rounded to 2 decimal places).$

Final Answers

• a) $\varepsilon(p) = \frac{2(p + 11)p}{480 - (p + 11)^2}$
• b) $\varepsilon(3) \approx 0.30$

Would you like a deeper explanation of any step?
Here are 5 questions for further exploration:

1. What does elasticity represent in economics?
2. How is price elasticity interpreted for $\varepsilon(p) > 1$, $\varepsilon(p) = 1$, and $\varepsilon(p) < 1$?
3. How can we use elasticity to predict changes in revenue?
4. How does the quadratic demand function affect elasticity compared to linear demand functions?
5. What implications does an elasticity of 0.30 have for the producer’s pricing strategy?

Tip: Always check the units and scaling when calculating elasticity—it helps interpret the results accurately!