To determine the point elasticity of demand, we use the formula:

$E_d = \frac{dQ}{dP} \times \frac{P}{Q}$

Where:

• $E_d$ is the point elasticity of demand,
• $\frac{dQ}{dP}$ is the derivative of the quantity $Q$ with respect to price $P$,
• $P$ is the price per unit, and
• $Q$ is the quantity demanded.

Step 1: Express $Q$ as a function of $P$

Given the demand function $p = 300 - q^2$, we can solve for $q$ in terms of $p$: $p = 300 - q^2$ $q^2 = 300 - p$ $q = \sqrt{300 - p}$

Step 2: Calculate the derivative $\frac{dQ}{dP}$

Differentiate $q = \sqrt{300 - p}$ with respect to $p$: $\frac{dq}{dp} = \frac{d}{dp} \left( \sqrt{300 - p} \right) = \frac{-1}{2 \sqrt{300 - p}}$

Step 3: Determine the price $p$ when $q = 5$

Substitute $q = 5$ into the demand function $p = 300 - q^2$: $p = 300 - 5^2 = 300 - 25 = 275$

So, when $q = 5$, the price $p = 275$.

Step 4: Calculate the point elasticity of demand

Now that we have $\frac{dq}{dp} = \frac{-1}{2 \sqrt{300 - p}}$, substitute $p = 275$ into this derivative: $\frac{dq}{dp} = \frac{-1}{2 \sqrt{300 - 275}} = \frac{-1}{2 \sqrt{25}} = \frac{-1}{2 \times 5} = \frac{-1}{10}$

Finally, substitute $p = 275$ and $q = 5$ into the elasticity formula: $E_d = \left( \frac{-1}{10} \right) \times \frac{275}{5} = \frac{-1}{10} \times 55 = -5.5$

Conclusion:

The point elasticity of demand when $q = 5$ is $E_d = -5.5$.

Let me know if you would like further details or if anything is unclear.

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Here are some related questions you might find interesting:

1. How would the elasticity change if the demand function was linear, like $p = 300 - 2q$?
2. How does the price elasticity of demand affect pricing strategies for a firm?
3. How does elasticity of demand differ between elastic, inelastic, and unitary demand?
4. What are the implications of a negative elasticity value for pricing and sales?
5. How would we calculate the total revenue at $q = 5$ and its relationship with elasticity?

Tip: Elasticity can help firms determine whether to increase or decrease prices to maximize revenue based on the sensitivity of quantity demanded to price changes.