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.