The point of unit elasticity occurs when the price elasticity of demand (E) is equal to 1. The formula for price elasticity of demand is:

$E = -\frac{p}{D(p)} \cdot D'(p)$

Here, the demand function is:

$D(p) = -0.01p + 6.45$

Step 1: Compute the derivative of $D(p)$

$D'(p) = -0.01$

Step 2: Set up the equation for unit elasticity

For unit elasticity, $E = 1$, so:

$1 = -\frac{p}{D(p)} \cdot D'(p)$

Substitute $D'(p) = -0.01$:

$1 = -\frac{p}{-0.01p + 6.45} \cdot (-0.01)$

Simplify the equation:

$1 = \frac{0.01p}{-0.01p + 6.45}$

Step 3: Solve for $p$

Multiply through by $-0.01p + 6.45$ (assuming it is not zero):

$-0.01p + 6.45 = 0.01p$

Combine terms:

$6.45 = 0.02p$

Solve for $p$:

$p = \frac{6.45}{0.02} = 322.5$

Step 4: Find $D(p)$ at $p = 322.5$

Substitute $p = 322.5$ into $D(p)$:

$D(322.5) = -0.01(322.5) + 6.45$

$D(322.5) = -3.225 + 6.45 = 3.225 \, \text{million chairs}$

Final Answer:

The point of unit elasticity occurs when: $p = 322.5 \, \text{dollars}$ and $D(p) = 3.225 \, \text{million chairs}.$

Let me know if you would like detailed explanations for any of these steps. Here are some related questions to expand understanding:

1. How is price elasticity of demand calculated in general?
2. What happens to revenue when the price is set at the point of unit elasticity?
3. What does the sign of $D'(p)$ indicate about the demand curve?
4. How would the elasticity change if the demand function had a different slope?
5. Can elasticity ever be negative, and what would that imply?

Tip: The concept of elasticity is crucial in pricing strategy. Pricing at unit elasticity maximizes revenue.