The demand function for a certain item is x=120(180-p squared) . Evaluate the elasticity at 4. Then for what value of p is the demand unit elastic?

To solve this problem, we need to calculate the **elasticity of demand** \( E(p) \), and then evaluate it at \( p = 4 \). The formula for elasticity of demand is:

\[
E(p) = -\frac{p}{x} \cdot \frac{dx}{dp}
\]

### Step 1: Express \( x \) as a function of \( p \)

The given demand function is:

\[
x = 120(180 - p^2)
\]

### Step 2: Differentiate \( x \) with respect to \( p \)

We differentiate \( x \) with respect to \( p \):

\[
\frac{dx}{dp} = 120 \cdot (-2p) = -240p
\]

### Step 3: Substitute into the elasticity formula

The elasticity formula is:

\[
E(p) = -\frac{p}{x} \cdot \frac{dx}{dp}
\]

Substituting \( \frac{dx}{dp} = -240p \) and \( x = 120(180 - p^2) \), we get:

\[
E(p) = -\frac{p}{120(180 - p^2)} \cdot (-240p)
\]

This simplifies to:

\[
E(p) = \frac{240p^2}{120(180 - p^2)} = \frac{2p^2}{180 - p^2}
\]

### Step 4: Evaluate at \( p = 4 \)

Substitute \( p = 4 \) into the elasticity formula:

\[
E(4) = \frac{2(4^2)}{180 - 4^2} = \frac{2(16)}{180 - 16} = \frac{32}{164} = \frac{8}{41}
\]

So, the elasticity at \( p = 4 \) is \( \frac{8}{41} \).

### Step 5: Find when demand is unit elastic

For demand to be **unit elastic**, the elasticity must equal 1:

\[
\frac{2p^2}{180 - p^2} = 1
\]

Solve this equation:

\[
2p^2 = 180 - p^2
\]

\[
3p^2 = 180
\]

\[
p^2 = 60
\]

\[
p = \sqrt{60} \approx 7.746
\]

Thus, the demand is unit elastic when \( p \approx 7.746 \).

### Summary:
- The elasticity at \( p = 4 \) is \( \frac{8}{41} \).
- The demand is unit elastic when \( p \approx 7.746 \).

Would you like more details or have any questions?

Here are 5 related questions to further explore this topic:
1. What does it mean for demand to be unit elastic?
2. How does elasticity affect pricing strategies?
3. What happens to revenue when demand is elastic or inelastic?
4. Can you find the price elasticity for different values of \( p \)?
5. How is elasticity used in real-world economics?

**Tip:** A price elasticity less than 1 (inelastic demand) means that the quantity demanded changes less than the price, while elasticity greater than 1 (elastic demand) indicates a larger change in demand for a price change.

The demand function for a certain item is x = 120(180 - p^2). Evaluate the elasticity at 4. Then for what value of p is the demand unit elastic?

Learn how to calculate the elasticity of demand for the function x = 120(180 - p^2) and determine when demand is unit elastic. This problem involves using differentiation and algebraic manipulation to solve for elasticity at a given price point. Suitable for advanced high school and college students.

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