The point elasticity of demand is given by the formula:

$E(p) = \left( \frac{dQ}{dP} \right) \times \left( \frac{P}{Q} \right)$

However, we have the demand function in terms of $P$ and $Q$, so we first need to express the derivative of $Q$ with respect to $P$ (i.e., $\frac{dQ}{dP}$) from the given demand equation. The demand function is:

$P = \frac{400}{Q} + 2$

Step 1: Solve for $Q$ in terms of $P$.

Rearranging the equation to solve for $Q$:

$P - 2 = \frac{400}{Q}$

$Q = \frac{400}{P - 2}$

Step 2: Compute $\frac{dQ}{dP}$.

Differentiating $Q = \frac{400}{P - 2}$ with respect to $P$:

$\frac{dQ}{dP} = -\frac{400}{(P - 2)^2}$

Step 3: Find the price $P$ when $Q = 100$.

Using the demand equation $P = \frac{400}{Q} + 2$, substitute $Q = 100$:

$P = \frac{400}{100} + 2 = 4 + 2 = 6$

Step 4: Calculate the point elasticity at $Q = 100$.

Now that we have $P = 6$ and $Q = 100$, substitute these values into the elasticity formula. We already know that:

$\frac{dQ}{dP} = -\frac{400}{(P - 2)^2}$

Substitute $P = 6$:

$\frac{dQ}{dP} = -\frac{400}{(6 - 2)^2} = -\frac{400}{16} = -25$

Now substitute into the elasticity formula:

$E(p) = \left( -25 \right) \times \left( \frac{6}{100} \right) = -25 \times 0.06 = -1.5$

Final Answer:

The point elasticity of demand when $Q = 100$ is $-1.5$.

Let me know if you'd like any further details or if you have additional questions!

---

Related Questions:

1. How would the point elasticity of demand change if the price were increased to 8?
2. What is the interpretation of a price elasticity of -1.5?
3. How can we calculate the total revenue based on the point elasticity of demand?
4. What would happen to demand if the elasticity were greater than 1?
5. How can we derive the inverse demand function from a given demand function?

Tip: When calculating price elasticity of demand, the sign of the elasticity is always negative because of the inverse relationship between price and quantity demanded (law of demand). However, we typically focus on the magnitude of the elasticity.