To solve this problem, let's go through the steps one by one.

Given Information:

The demand function is: $q = D(x) = \frac{300}{x}$

Part (a): Finding the Equation for Elasticity, $E(x)$

The elasticity of demand is defined as: $E(x) = -\frac{dD(x)}{dx} \cdot \frac{x}{D(x)}$

Step 1: Find $D(x)$ and $\frac{dD(x)}{dx}$.

1. Rewrite the demand function: $D(x) = \frac{300}{x}$

2. Differentiate $D(x)$ with respect to $x$: $\frac{dD(x)}{dx} = -\frac{300}{x^2}$

Step 2: Substitute into the Elasticity Formula.

Now, we substitute $D(x)$ and $\frac{dD(x)}{dx}$ into the elasticity formula: $E(x) = -\left(-\frac{300}{x^2}\right) \cdot \frac{x}{\frac{300}{x}}$

Simplify this expression:

1. The negative signs cancel out: $E(x) = \frac{300}{x^2} \cdot \frac{x}{\frac{300}{x}}$

2. Multiply the terms: $E(x) = \frac{300x}{x^2 \cdot \frac{300}{x}} = \frac{x}{x} = 1$

So, the elasticity $E(x)$ is:

$E(x) = 1$

This is the elasticity function, and it tells us about the elasticity of the demand at any price level $x$.

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Would you like further details on parts (b) and (c), or have questions about this solution?

Expanding Questions

1. How does elasticity impact revenue?
2. What determines whether demand is elastic, inelastic, or unit elastic?
3. Why is differentiation used to find elasticity?
4. How does price elasticity of demand vary with different types of products?
5. What is the economic significance of unit elasticity?

Tip:

For revenue maximization, remember that demand elasticity is a key factor. When demand is unit elastic, revenue is typically maximized at that price point.