To determine the level of production $x$ for which the demand is elastic, we need to find where the elasticity of demand $E(x)$ is greater than 1.

Step 1: Understand the Demand Function

The demand function is given by: $p = D(x) = 168 - 14\sqrt{x}$

Step 2: Compute $\frac{dp}{dx}$

To find the elasticity, we need to calculate $\frac{dp}{dx}$:

1. Rewrite $p = 168 - 14x^{1/2}$.
2. Differentiate $p$ with respect to $x$: $\frac{dp}{dx} = -\frac{14}{2}x^{-\frac{1}{2}} = -\frac{7}{\sqrt{x}}$

Step 3: Compute the Elasticity of Demand $E(x)$

The elasticity of demand formula is: $E(x) = -\frac{dp}{dx} \cdot \frac{x}{p}$ Substitute $\frac{dp}{dx} = -\frac{7}{\sqrt{x}}$ and $p = 168 - 14\sqrt{x}$: $E(x) = -\left(-\frac{7}{\sqrt{x}}\right) \cdot \frac{x}{168 - 14\sqrt{x}}$ $E(x) = \frac{7x^{1/2}}{168 - 14x^{1/2}}$

Step 4: Set $E(x) > 1$ to Find When Demand is Elastic

We need to solve for $x$ such that: $\frac{7x^{1/2}}{168 - 14x^{1/2}} > 1$ Multiply both sides by $168 - 14x^{1/2}$: $7x^{1/2} > 168 - 14x^{1/2}$ Combine like terms: $21x^{1/2} > 168$ $x^{1/2} > 8$ Square both sides: $x > 64$

Conclusion

The demand is elastic when $x > 64$. Since only whole units can be produced, the minimum production level for which the demand is elastic is $x = 65$.

Would you like a further breakdown of any steps or have questions on a specific part?

Here are some related questions to explore further:

1. What is the relationship between demand elasticity and revenue?
2. How does elasticity affect production decisions in economics?
3. Can the elasticity of a product change over time? If so, why?
4. What happens to the price when the demand is elastic versus inelastic?
5. How is the elasticity of demand influenced by substitutes?

Tip: Elastic demand indicates that a change in quantity will significantly impact price, which is critical in pricing strategy.