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To solve the given problem:

### Part a: Find the price elasticity of demand \( \varepsilon \) when \( p = 12 \).

The formula for the price elasticity of demand \( \varepsilon \) is:

\[
\varepsilon = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right|
\]

We are given the demand function:

\[
q = \frac{1}{3}(300 - p^2)
\]

First, we need to compute the derivative \( \frac{dq}{dp} \).

### Step 1: Differentiate \( q \) with respect to \( p \):

\[
q = \frac{1}{3}(300 - p^2)
\]
\[
\frac{dq}{dp} = \frac{1}{3} \cdot (-2p) = -\frac{2p}{3}
\]

### Step 2: Find \( q \) when \( p = 12 \):

Substitute \( p = 12 \) into the original demand function:

\[
q = \frac{1}{3}(300 - 12^2) = \frac{1}{3}(300 - 144) = \frac{1}{3}(156) = 52
\]

### Step 3: Calculate \( \varepsilon \) at \( p = 12 \):

Using the elasticity formula:
\[
\varepsilon = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right|
\]

Substitute the values \( p = 12 \), \( q = 52 \), and \( \frac{dq}{dp} = -\frac{2p}{3} = -\frac{2(12)}{3} = -8 \):

\[
\varepsilon = \left| \frac{12}{52} \cdot (-8) \right| = \left| \frac{12 \cdot (-8)}{52} \right| = \left| \frac{-96}{52} \right| = \frac{96}{52} = \frac{24}{13} \approx 1.846
\]

### Part b: Is the demand elastic, inelastic, or unit elastic?

If \( \varepsilon > 1 \), demand is **elastic**. If \( \varepsilon = 1 \), demand is **unit elastic**. If \( \varepsilon < 1 \), demand is **inelastic**.

Since \( \varepsilon \approx 1.846 \), the demand is **elastic**.

---

Would you like more details on this process or have any further questions? Here are some related questions:

1. What is the economic interpretation of elastic demand?
2. How does the price elasticity of demand affect total revenue?
3. What would happen to elasticity if the price increased to \( p = 15 \)?
4. How is elasticity related to the slope of the demand curve?
5. What are other factors that can influence the elasticity of demand?

**Tip**: Elasticity is a measure of responsiveness. For elastic demand, consumers are highly responsive to price changes!

For a particular commodity, the demand function is q = (1/3)(300 - p^2). a) Find the price elasticity of demand (ε) when p = 12. b) Is demand elastic, inelastic, or unit elastic?

This problem calculates the price elasticity of demand using a given demand function q = (1/3)(300 - p^2). Find ε when p = 12 and determine if the demand is elastic, inelastic, or unit elastic. This solution involves differentiation and key economic concepts.

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