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The given price-demand equation is \( p = 9000 - 0.1x^2 \), where \( p \) is the price and \( x \) is the demand.

We are asked to find the values of \( x \) where the demand is elastic or inelastic, using the elasticity of demand formula:

\[
E(x) = - \frac{g(x)}{xg'(x)}
\]

where \( g(x) \) is the price function and \( g'(x) \) is its derivative with respect to \( x \).

### Step 1: Find \( g'(x) \)
The price-demand equation is:
\[
g(x) = 9000 - 0.1x^2
\]

Now, differentiate this equation with respect to \( x \):
\[
g'(x) = \frac{d}{dx} \left( 9000 - 0.1x^2 \right) = -0.2x
\]

### Step 2: Substitute into the elasticity of demand formula
Now, substitute \( g(x) = 9000 - 0.1x^2 \) and \( g'(x) = -0.2x \) into the elasticity formula:

\[
E(x) = - \frac{9000 - 0.1x^2}{x(-0.2x)}
\]

Simplify the denominator:
\[
E(x) = - \frac{9000 - 0.1x^2}{-0.2x^2}
\]
\[
E(x) = \frac{9000 - 0.1x^2}{0.2x^2}
\]

Now, split the fraction:
\[
E(x) = \frac{9000}{0.2x^2} - \frac{0.1x^2}{0.2x^2}
\]
\[
E(x) = \frac{9000}{0.2x^2} - 0.5
\]
\[
E(x) = \frac{45000}{x^2} - 0.5
\]

### Step 3: Analyze elasticity
To determine whether demand is elastic or inelastic, we need to analyze the value of \( E(x) \):
- **Elastic demand**: \( E(x) > 1 \)
- **Inelastic demand**: \( E(x) < 1 \)

Thus, we need to solve the inequality:
\[
\frac{45000}{x^2} - 0.5 > 1
\]
\[
\frac{45000}{x^2} > 1.5
\]
\[
x^2 < \frac{45000}{1.5} = 30000
\]
\[
x < \sqrt{30000} \approx 173.21
\]

So, demand is **elastic** when \( x < 173.21 \).

For inelastic demand:
\[
\frac{45000}{x^2} - 0.5 < 1
\]
\[
\frac{45000}{x^2} < 1.5
\]
\[
x^2 > 30000
\]
\[
x > 173.21
\]

Thus, demand is **inelastic** when \( x > 173.21 \).

### Conclusion
- Demand is elastic for \( x < 173.21 \).
- Demand is inelastic for \( x > 173.21 \).

Would you like further details on any of these steps, or have any questions?

Here are 5 related questions for further exploration:
1. How does the concept of unit elasticity apply to this problem?
2. What would happen to elasticity if the price-demand function changed to a linear one?
3. How would the elasticity equation change if a constant price was added?
4. Can you derive the point of unit elasticity from the general equation?
5. How does elasticity relate to revenue maximization in economic models?

**Tip:** Elasticity greater than 1 suggests that consumers are sensitive to price changes, while values less than 1 suggest insensitivity.

If a price-demand equation is solved for p, then price is expressed as p = g(x) and x becomes the independent variable. Find the values of x for which demand is elastic and for which demand is inelastic. The price-demand equation is p = 9000 - 0.1x^2.

Learn how to analyze the elasticity of demand using the price-demand equation p = 9000 - 0.1x². This detailed solution involves differentiation and solving inequalities to find the points where demand is elastic or inelastic. Suitable for advanced economics and calculus students in Grades 11-12.

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