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?

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.