https://file.aiquickdraw.com/mathbot/file/1729784936040yvabhmdp.jpg

Let's analyze the given problem and find the solutions.

### Problem Overview:
The demand function is given as:
\[
q = \frac{80,000}{p}
\]
where \( q \) is the quantity demanded, and \( p \) is the price.

#### (a) Find \( \varepsilon \) when \( p = 8 \).

The price elasticity of demand \( \varepsilon \) is defined as:
\[
\varepsilon = \frac{p}{q} \cdot \frac{dq}{dp}
\]
We first need to find the derivative \( \frac{dq}{dp} \).

1. From the demand function \( q = \frac{80,000}{p} \), differentiate \( q \) with respect to \( p \):
\[
\frac{dq}{dp} = -\frac{80,000}{p^2}
\]

2. Now, substitute \( p = 8 \) into the equation and calculate \( q \) at \( p = 8 \):
\[
q = \frac{80,000}{8} = 10,000
\]

3. Substituting \( p = 8 \), \( q = 10,000 \), and \( \frac{dq}{dp} = -\frac{80,000}{64} = -1,250 \) into the elasticity formula:
\[
\varepsilon = \frac{8}{10,000} \cdot (-1,250) = -1
\]

Thus, \( \varepsilon = -1 \).

#### (b) Is the demand elastic, inelastic, or unit elastic?

Since \( \varepsilon = -1 \), the demand is **unit elastic**. In this case, a 1% change in price results in a 1% change in quantity demanded, so total revenue remains constant.

---

Would you like any further details or explanations?

### 5 Related Questions:
1. How does elasticity affect total revenue when demand is elastic?
2. What happens to total revenue when demand is inelastic?
3. How does elasticity change at different points along a linear demand curve?
4. What is the relationship between marginal revenue and price elasticity of demand?
5. How does the elasticity of demand vary between different types of goods?

### Tip:
Remember that when \( |\varepsilon| > 1 \), demand is elastic, meaning consumers are very responsive to price changes. When \( |\varepsilon| < 1 \), demand is inelastic, and consumers are less responsive to price changes.

For a particular commodity, the demand function is q = 80,000 / p. (a) Find ε when p = 8. (b) Is demand elastic, inelastic, or unit elastic?

Learn how to calculate the price elasticity of demand for the given demand function q = 80,000 / p. Find the elasticity when p = 8 and determine whether the demand is elastic, inelastic, or unit elastic.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation