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To solve this problem, we need to calculate the elasticity of demand \( E(p) \) using the given price-demand function:

\[
x = 42,336 - 32p^2
\]

### Step 1: Derive the Elasticity of Demand Formula
The elasticity of demand \( E(p) \) is defined as:

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

where:
- \( p \) is the price,
- \( x \) is the demand function (in terms of \( p \)),
- \( \frac{dx}{dp} \) is the derivative of \( x \) with respect to \( p \).

### Step 2: Differentiate \( x \) with respect to \( p \)
Given:
\[
x = 42,336 - 32p^2
\]

Let's differentiate \( x \) with respect to \( p \):
\[
\frac{dx}{dp} = -64p
\]

### Step 3: Substitute and Calculate \( E(p) \) for Each Value of \( p \)

For each case, we will substitute the specific value of \( p \) into the elasticity formula.

#### (a) When \( p = 18 \):
1. Substitute \( p = 18 \) into the demand function to find \( x \):
   \[
   x = 42,336 - 32 \times 18^2 = 42,336 - 32 \times 324 = 42,336 - 10,368 = 31,968
   \]

2. Substitute \( p = 18 \), \( x = 31,968 \), and \( \frac{dx}{dp} = -64 \times 18 = -1152 \) into \( E(p) \):
   \[
   E(18) = \frac{18}{31,968} \times (-1152) = \frac{18 \times -1152}{31,968} = -0.648
   \]

Since \( |E(18)| < 1 \), demand is **inelastic** at \( p = 18 \).

#### (b) When \( p = 29 \):
1. Substitute \( p = 29 \) into the demand function to find \( x \):
   \[
   x = 42,336 - 32 \times 29^2 = 42,336 - 32 \times 841 = 42,336 - 26,912 = 15,424
   \]

2. Substitute \( p = 29 \), \( x = 15,424 \), and \( \frac{dx}{dp} = -64 \times 29 = -1856 \) into \( E(p) \):
   \[
   E(29) = \frac{29}{15,424} \times (-1856) = \frac{29 \times -1856}{15,424} = -3.49
   \]

Since \( |E(29)| > 1 \), demand is **elastic** at \( p = 29 \).

#### (c) When \( p = 20 \):
1. Substitute \( p = 20 \) into the demand function to find \( x \):
   \[
   x = 42,336 - 32 \times 20^2 = 42,336 - 32 \times 400 = 42,336 - 12,800 = 29,536
   \]

2. Substitute \( p = 20 \), \( x = 29,536 \), and \( \frac{dx}{dp} = -64 \times 20 = -1280 \) into \( E(p) \):
   \[
   E(20) = \frac{20}{29,536} \times (-1280) = \frac{20 \times -1280}{29,536} = -0.87
   \]

Since \( |E(20)| < 1 \), demand is **inelastic** at \( p = 20 \).

### Summary of Results
1. \( p = 18 \): \( E(18) = -0.65 \) (Inelastic)
2. \( p = 29 \): \( E(29) = -3.49 \) (Elastic)
3. \( p = 20 \): \( E(20) = -0.87 \) (Inelastic)

Would you like further details or have any questions?

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#### Related Questions
1. What is the significance of elasticity being less than, equal to, or greater than 1?
2. How does elasticity affect pricing decisions for a business?
3. Why is the demand function often modeled as quadratic or other polynomial forms?
4. How would changes in consumer income affect the elasticity of demand?
5. Can elasticity of demand vary over different ranges of prices?

#### Tip
Elasticity can help businesses understand how sensitive consumers are to price changes, guiding pricing strategy to maximize revenue.

Given the following price-demand function, find the elasticity of demand, E(p), and determine whether demand is elastic, inelastic, or has unit elasticity for the following values of p (rounded to two decimal places): p = 18, p = 29, and p = 20.

Learn how to calculate the elasticity of demand using the price-demand function x = 42,336 - 32p^2. This problem covers the differentiation of demand functions and elasticity calculations for values of p = 18, p = 29, and p = 20, helping to determine whether demand is elastic, inelastic, or has unit elasticity.

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