To find the point elasticity of demand for the given equation $p = \frac{20}{q^2}$ at $q = 10$, we will use the formula for point elasticity of demand:

$E_d = \left( \frac{p}{q} \right) \cdot \frac{dQ}{dp}$

First, we need to follow these steps:

Step 1: Find $p$ when $q = 10$

From the demand equation, $p = \frac{20}{q^2}$. Substituting $q = 10$:

$p = \frac{20}{10^2} = \frac{20}{100} = 0.2$

So, $p = 0.2$ when $q = 10$.

Step 2: Find $\frac{dp}{dq}$

To compute the point elasticity, we also need to find the derivative of $p$ with respect to $q$, which is $\frac{dp}{dq}$.

We start with the demand function:

$p = \frac{20}{q^2}$

Taking the derivative with respect to $q$:

$\frac{dp}{dq} = -\frac{40}{q^3}$

Step 3: Find the value of $\frac{dp}{dq}$ when $q = 10$

Now, substitute $q = 10$ into the derivative:

$\frac{dp}{dq} = -\frac{40}{10^3} = -\frac{40}{1000} = -0.04$

Step 4: Calculate the point elasticity of demand

Now, we can calculate the point elasticity of demand at $q = 10$ using the formula:

$E_d = \left( \frac{p}{q} \right) \cdot \frac{dp}{dq}$

Substitute $p = 0.2$, $q = 10$, and $\frac{dp}{dq} = -0.04$:

$E_d = \left( \frac{0.2}{10} \right) \cdot (-0.04) = 0.02 \cdot (-0.04) = -0.0008$

Thus, the point elasticity of demand at $q = 10$ is $-0.0008$.

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

Further Exploration:

1. How would the elasticity change if we used a different demand function, such as $p = 30/q$?
2. How can the concept of elasticity of demand be applied to pricing strategies for firms?
3. How does elasticity relate to revenue and how can it help firms make optimal decisions?
4. What does the sign of the elasticity value tell us about the nature of demand?
5. How does the formula for elasticity change when dealing with linear demand functions?

Tip: Elasticity is a crucial concept in economics because it shows how sensitive the quantity demanded is to changes in price, which can help businesses set prices more effectively.