To solve this problem, we need to analyze the demand function using the concept of elasticity of demand and examine its behavior with respect to revenue. Let's go step by step.

Part (a): Elasticity of Demand

We are given the demand function: $p = 880 - x, \quad 0 \leq x \leq 880$ where $p$ is the price per unit (in dollars) and $x$ is the number of units.

Elasticity of Demand Formula:

The elasticity of demand $E(x)$ is given by: $E(x) = \left| \frac{d(x) \cdot p(x)}{R'(x)} \right| = \left| \frac{x \cdot p'(x)}{p(x)} \right|$

Here,

• $p(x)$ is the price function $p(x) = 880 - x$.
• $p'(x)$ is the derivative of the price function, which is: $p'(x) = \frac{d}{dx}(880 - x) = -1$

Thus, the elasticity of demand becomes: $E(x) = \left| \frac{x \cdot (-1)}{880 - x} \right| = \frac{x}{880 - x}$

When is Demand Elastic, Inelastic, or Unit Elastic?

• Elastic demand: when $E(x) > 1$.
• Inelastic demand: when $E(x) < 1$.
• Unit elasticity: when $E(x) = 1$.

Elastic Demand ( $E(x) > 1$ ):

To find when the demand is elastic: $\frac{x}{880 - x} > 1$ Solving this inequality: $x > 880 - x$ $2x > 880 \quad \Rightarrow \quad x > 440$ Thus, the demand is elastic when $x \in (440, 880)$.

Inelastic Demand ( $E(x) < 1$ ):

For inelastic demand: $\frac{x}{880 - x} < 1$ Solving this inequality: $x < 880 - x$ $2x < 880 \quad \Rightarrow \quad x < 440$ Thus, the demand is inelastic when $x \in (0, 440)$.

Unit Elasticity ( $E(x) = 1$ ):

To find when the demand is unit elastic: $\frac{x}{880 - x} = 1$ Solving this equation: $x = 880 - x$ $2x = 880 \quad \Rightarrow \quad x = 440$ So, the demand is of unit elasticity when $x = 440$.

Summary for Part (a):

• Elastic demand: $(440, 880)$
• Inelastic demand: $(0, 440)$
• Unit elasticity: $x = 440$

Part (b): Revenue Function Behavior

The revenue function $R(x)$ is given by: $R(x) = x \cdot p(x) = x \cdot (880 - x) = 880x - x^2$

To describe the behavior of the revenue function:

1. The revenue function is a quadratic function $R(x) = 880x - x^2$, which is a parabola that opens downwards (since the coefficient of $x^2$ is negative).
2. The maximum point of this parabola occurs at $x = 440$, which is the point where demand is unit elastic (as found in part (a)).
3. Thus, the revenue increases as $x$ approaches 440, then flattens at $x = 440$, and decreases for $x > 440$.

Summary for Part (b):

• The revenue function increases for $0 \leq x < 440$, is flat at $x = 440$, and decreases for $x > 440$.

Would you like more details on any part of this solution, or do you have any additional questions?

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Here are 5 related questions to expand your understanding:

1. What happens to the price when the quantity demanded increases?
2. How does the concept of marginal revenue relate to elasticity of demand?
3. How can the elasticity of demand affect pricing strategies in real-world markets?
4. What is the significance of unit elasticity in business decision-making?
5. How does the elasticity of demand change if the demand curve shifts?

Tip: Elasticity of demand is crucial for understanding how sensitive consumers are to price changes. It helps businesses set prices optimally for maximum revenue!