The demand function for a product is modeled by
p = 880 − x,    0 ≤ x ≤ 880
where p is the price per unit (in dollars) and x is the number of units.
(a)
Determine when the demand is elastic and inelastic. (Enter your answer using interval notation.)
elastic demand	
  
inelastic demand

The demand function is given as \( p = 880 - x \), where \( p \) is the price per unit and \( x \) is the number of units. To determine when the demand is elastic or inelastic, we need to use the concept of **elasticity of demand**.

### Elasticity of demand formula:
The elasticity of demand, \( E(x) \), is given by:

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

where:
- \( p(x) \) is the price function,
- \( p'(x) \) is the derivative of the price function with respect to \( x \).

### Step 1: Find the derivative of the price function.
Given \( p(x) = 880 - x \), the derivative is:

\[
p'(x) = -1
\]

### Step 2: Calculate the elasticity of demand.
Substitute \( p'(x) \) and \( p(x) \) into the elasticity formula:

\[
E(x) = -\frac{-1 \cdot x}{880 - x} = \frac{x}{880 - x}
\]

### Step 3: Determine when the demand is elastic or inelastic.
- **Elastic demand**: This occurs when \( E(x) > 1 \) (percentage change in quantity demanded is greater than the percentage change in price).
- **Inelastic demand**: This occurs when \( E(x) < 1 \) (percentage change in quantity demanded is less than the percentage change in price).
- **Unitary elastic demand**: This occurs when \( E(x) = 1 \).

Set the condition for elasticity and solve for \( x \).

#### Elastic demand \( E(x) > 1 \):
\[
\frac{x}{880 - x} > 1
\]
Solving this inequality:
\[
x > 880 - x
\]
\[
2x > 880
\]
\[
x > 440
\]
Thus, the demand is elastic when \( x > 440 \).

#### Inelastic demand \( E(x) < 1 \):
\[
\frac{x}{880 - x} < 1
\]
Solving this inequality:
\[
x < 880 - x
\]
\[
2x < 880
\]
\[
x < 440
\]
Thus, the demand is inelastic when \( x < 440 \).

#### Unitary elasticity \( E(x) = 1 \):
At \( x = 440 \), the demand is unitary elastic.

### Final Answer:
- **Elastic demand**: \( (440, 880) \)
- **Inelastic demand**: \( (0, 440) \)

Would you like more details on how this elasticity is interpreted? 

Here are 5 related questions for you:
1. How does a business use elasticity of demand to set prices?
2. What happens to revenue when demand is elastic and prices are increased?
3. Can demand elasticity change over time for a product?
4. What factors affect whether a product has elastic or inelastic demand?
5. How would a shift in the demand curve impact elasticity?

**Tip**: Remember that elastic demand means consumers are sensitive to price changes, while inelastic demand means they are less sensitive.

The demand function for a product is modeled by p = 880 − x, 0 ≤ x ≤ 880, where p is the price per unit (in dollars) and x is the number of units. Determine when the demand is elastic and inelastic.

Learn how to determine when the demand is elastic or inelastic for the function p = 880 − x, where p is the price per unit and x is the number of units. This solution involves the elasticity of demand formula, the price function, and solving inequalities. Suitable for Grades 11-12 and undergraduate students.

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