Consider the following demand function:

p=D(x)=153−1.5x‾‾‾‾‾‾‾‾‾‾√
Step 1 of 2 :  Find the elasticity function.

To find the elasticity function, we will calculate the price elasticity of demand, denoted as \( E(x) \). The formula for the elasticity of demand function is:

\[
E(x) = \frac{dD(x)/dx}{D(x)/x} \cdot x
\]

Given that \( D(x) = 153 - 1.5\sqrt{x} \), let's find \( E(x) \) step by step.

### Step 1: Differentiate \( D(x) \) with respect to \( x \)

1. First, rewrite \( D(x) \) as:
   \[
   D(x) = 153 - 1.5x^{1/2}
   \]

2. Differentiate \( D(x) \) with respect to \( x \):
   \[
   \frac{dD(x)}{dx} = -1.5 \cdot \frac{1}{2} x^{-\frac{1}{2}} = -\frac{0.75}{\sqrt{x}}
   \]

### Step 2: Set up the Elasticity Formula

The elasticity function is:
\[
E(x) = \frac{\frac{dD(x)}{dx}}{\frac{D(x)}{x}} \cdot x
\]

Now, substitute \( \frac{dD(x)}{dx} = -\frac{0.75}{\sqrt{x}} \) and \( D(x) = 153 - 1.5\sqrt{x} \) into the formula.

### Step 3: Simplify \( E(x) \)

1. Calculate \( \frac{D(x)}{x} \):
   \[
   \frac{D(x)}{x} = \frac{153 - 1.5\sqrt{x}}{x}
   \]

2. Now, substitute these into the elasticity formula:
   \[
   E(x) = \frac{-\frac{0.75}{\sqrt{x}}}{\frac{153 - 1.5\sqrt{x}}{x}} \cdot x
   \]

3. Simplify the expression to find \( E(x) \):
   \[
   E(x) = -\frac{0.75x}{\sqrt{x}(153 - 1.5\sqrt{x})}
   \]

4. Simplify further by cancelling terms as possible.

Let me know if you'd like to see further simplification, or if you have any other questions!

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Here are five questions that could help deepen understanding of this topic:

1. How does the elasticity of demand change as \( x \) increases?
2. What does it mean if the elasticity \( E(x) \) is greater than 1, less than 1, or equal to 1?
3. How does the square root in the demand function affect the elasticity?
4. What are the interpretations of the elasticity of demand in terms of revenue?
5. How would the elasticity function change if the demand function were linear?

**Tip:** Remember that elasticity helps determine how sensitive quantity demanded is to changes in price. This sensitivity can be crucial for setting prices in a way that maximizes revenue or market share.

Consider the following demand function: p = D(x) = 153 − 1.5√x. Step 1 of 2: Find the elasticity function.

Explore a step-by-step solution to find the elasticity function of the demand equation p = D(x) = 153 − 1.5√x. This problem involves key calculus concepts like differentiation and elasticity of demand, commonly used in economics. Suitable for undergraduate economics or calculus students.

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