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### Problem Analysis

We are given a linear demand equation for a commodity:
\[
q = 240 - 2p \quad \text{for } 0 \leq p \leq 120.
\]

The questions are as follows:
1. **(a)** Express the elasticity of demand as a function of \( p \).
2. **(b)** Calculate the elasticity of demand when the price is \( p = 100 \). Interpret your answer.
3. **(c)** Calculate the elasticity of demand when the price is \( p = 50 \). Interpret your answer.
4. **(d)** At what price is the elasticity of demand equal to \( -1 \)? What is the economic significance of this price?

### Solution Steps

#### (a) Elasticity of Demand as a Function of \( p \)

Elasticity of demand \( E(p) \) is defined as:
\[
E(p) = \frac{p}{q} \cdot \frac{dq}{dp},
\]
where:
- \( q = 240 - 2p \) (the demand function),
- \( \frac{dq}{dp} \) is the derivative of \( q \) with respect to \( p \).

Calculate \( \frac{dq}{dp} \):
\[
\frac{dq}{dp} = -2.
\]

Now substitute \( q \) and \( \frac{dq}{dp} \) into the elasticity formula:
\[
E(p) = \frac{p}{240 - 2p} \cdot (-2) = \frac{-2p}{240 - 2p}.
\]

Thus, the elasticity of demand as a function of \( p \) is:
\[
E(p) = \frac{-2p}{240 - 2p}.
\]

#### (b) Elasticity of Demand When \( p = 100 \)

Substitute \( p = 100 \) into the elasticity function:
\[
E(100) = \frac{-2(100)}{240 - 2(100)} = \frac{-200}{240 - 200} = \frac{-200}{40} = -5.
\]

**Interpretation**: When \( p = 100 \), the elasticity of demand is \( -5 \). This means that a 1% increase in price leads to a 5% decrease in the quantity demanded. The demand is **elastic** at this price since the magnitude of elasticity is greater than 1.

#### (c) Elasticity of Demand When \( p = 50 \)

Substitute \( p = 50 \) into the elasticity function:
\[
E(50) = \frac{-2(50)}{240 - 2(50)} = \frac{-100}{240 - 100} = \frac{-100}{140} \approx -0.71.
\]

**Interpretation**: When \( p = 50 \), the elasticity of demand is approximately \( -0.71 \). This means that a 1% increase in price leads to a 0.71% decrease in the quantity demanded. The demand is **inelastic** at this price since the magnitude of elasticity is less than 1.

#### (d) Price at Which Elasticity is Equal to \( -1 \)

Set \( E(p) = -1 \) and solve for \( p \):
\[
\frac{-2p}{240 - 2p} = -1.
\]

Simplify:
\[
-2p = - (240 - 2p),
\]
\[
-2p = -240 + 2p,
\]
\[
4p = 240,
\]
\[
p = 60.
\]

**Economic Significance**: When the price is \( p = 60 \), the elasticity of demand is \( -1 \), indicating **unitary elasticity**. This means that a 1% change in price leads to an equal 1% change in quantity demanded, implying that total revenue remains unchanged when the price changes.

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If you have further questions or would like more detailed explanations, feel free to ask! Here are some related questions:

1. What does it mean for a good to have elastic vs. inelastic demand?
2. How does the concept of elasticity affect pricing decisions?
3. How can revenue be maximized using the concept of elasticity?
4. What factors influence the elasticity of demand for a product?
5. How does the elasticity of a necessity good compare to that of a luxury good?

**Tip**: Elasticity can be a powerful tool for businesses to understand how sensitive consumers are to changes in price, which can help in making informed pricing and marketing decisions.

Suppose the demand q and price p for a certain commodity are related by the linear equation q = 240 - 2p (for 0 ≤ p ≤ 120). a. Express the elasticity of demand as a function of p. b. Calculate the elasticity of demand when the price is p = 100. Interpret your answer. c. Calculate the elasticity of demand when the price is p = 50. Interpret your answer. d. At what price is the elasticity of demand equal to -1? What is the economic significance of this price?

This problem covers the calculation of the elasticity of demand for a commodity where the demand equation is q = 240 - 2p. It includes deriving the elasticity formula, solving for specific price points (p = 100 and p = 50), and interpreting economic significance.

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