To compute the elasticity of demand $E(p)$, we need to use the elasticity formula:

$E(p) = \left( \frac{p}{x} \right) \cdot \left( \frac{dx}{dp} \right)$

Where:

• $p$ is the price function in terms of $x$,
• $\frac{dx}{dp}$ is the derivative of $x$ with respect to $p$.

The demand function given is:

$p(x) = \sqrt{6} - 0.02x$

Step 1: Express $x$ in terms of $p$

We start by solving the demand function for $x$:

$p = \sqrt{6} - 0.02x$

Rearranging for $x$:

$x = \frac{\sqrt{6} - p}{0.02}$

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

Differentiate $x(p)$ with respect to $p$:

$\frac{dx}{dp} = \frac{d}{dp} \left( \frac{\sqrt{6} - p}{0.02} \right) = \frac{-1}{0.02} = -50$

Step 3: Plug into the Elasticity Formula

Now, substitute $\frac{dx}{dp} = -50$ into the elasticity formula:

$E(p) = \left( \frac{p}{x} \right) \cdot (-50)$

Step 4: Express $x$ in terms of $p$

From the previous expression for $x$:

$x = \frac{\sqrt{6} - p}{0.02}$

Step 5: Final Expression for Elasticity

Now substitute $x$ into the elasticity formula:

$E(p) = \left( \frac{p}{\frac{\sqrt{6} - p}{0.02}} \right) \cdot (-50)$

Simplify:

$E(p) = \left( \frac{0.02p}{\sqrt{6} - p} \right) \cdot (-50)$

$E(p) = \frac{-50 \cdot 0.02p}{\sqrt{6} - p}$

$E(p) = \frac{-p}{\frac{\sqrt{6} - p}{50}}$

Thus, the elasticity of demand $E(p)$ is:

$E(p) = \frac{-p}{\sqrt{6} - p}$

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Would you like me to provide further explanation or details? Here are some related questions you might find helpful:

1. What is the interpretation of the elasticity of demand in this context?
2. How does the value of $E(p)$ determine whether the demand is elastic or inelastic?
3. What happens to elasticity as the price increases?
4. How can we interpret the demand function geometrically?
5. What is the significance of elasticity in setting prices for a product?

Tip: Elasticity greater than 1 means the product is elastic, while elasticity less than 1 indicates inelastic demand.