To find the price elasticity of demand when the quantity $Q = 30$ for the demand function $Q = \sqrt{1200 - 2P}$, we will follow these steps:

1. Find the corresponding price $P$ when $Q = 30$: $30 = \sqrt{1200 - 2P}$ Squaring both sides: $900 = 1200 - 2P$ Rearranging gives: $2P = 1200 - 900 = 300 \implies P = 150$

2. Calculate the derivative of the demand function with respect to price: The demand function can be rewritten as: $P = \frac{1200 - Q^2}{2}$ Taking the derivative with respect to $Q$: $\frac{dP}{dQ} = -\frac{Q}{2}$

3. Evaluate $\frac{dP}{dQ}$ at $Q = 30$: $\frac{dP}{dQ} = -\frac{30}{2} = -15$

4. Use the point elasticity of demand formula: The price elasticity of demand (PED) is given by: $\text{PED} = \frac{dQ}{dP} \cdot \frac{P}{Q}$ To find $\frac{dQ}{dP}$, we take the reciprocal of $\frac{dP}{dQ}$: $\frac{dQ}{dP} = -\frac{1}{15}$

5. Substituting into the elasticity formula: Using $P = 150$ and $Q = 30$: $\text{PED} = -\frac{1}{15} \cdot \frac{150}{30} = -\frac{1}{15} \cdot 5 = -\frac{1}{3}$

6. Classifying the elasticity: Since $|\text{PED}| = \frac{1}{3} < 1$, the demand is classified as inelastic.

Interpretation:

An elasticity of $-\frac{1}{3}$ means that a 1% increase in price would lead to a 0.33% decrease in quantity demanded. Because the demand is inelastic, consumers are relatively unresponsive to price changes in this range, indicating that they will continue to buy similar quantities even if prices rise.

Further Exploration:

1. What factors might contribute to the inelastic nature of this demand?
2. How would changes in consumer income affect this demand function?
3. Can you explain the implications of inelastic demand for a business setting prices?
4. What happens to demand elasticity if price is decreased instead of increased?
5. How can elasticity help in making pricing decisions?

Tip: Always consider the context of the product when analyzing elasticity; necessities often have inelastic demand, while luxury goods tend to be more elastic.