The demand function for a certain make of exercise bicycle sold exclusively through cable television is
p = squreroot 6 − 0.02x
    (0 ≤ x ≤ 300)
where p is the unit price in hundreds of dollars and x is the quantity demanded/week. Compute the elasticity of demand E(p).
E(p) =

To compute the elasticity of demand \( E(p) \), we need to use the elasticity formula:

\[
E(p) = \left( \frac{p}{x} \right) \cdot \left( \frac{dx}{dp} \right)
\]

Where:
- \( p \) is the price function in terms of \( x \),
- \( \frac{dx}{dp} \) is the derivative of \( x \) with respect to \( p \).

The demand function given is:

\[
p(x) = \sqrt{6} - 0.02x
\]

### Step 1: Express \( x \) in terms of \( p \)
We start by solving the demand function for \( x \):

\[
p = \sqrt{6} - 0.02x
\]

Rearranging for \( x \):

\[
x = \frac{\sqrt{6} - p}{0.02}
\]

### Step 2: Compute \( \frac{dx}{dp} \)
Differentiate \( x(p) \) with respect to \( p \):

\[
\frac{dx}{dp} = \frac{d}{dp} \left( \frac{\sqrt{6} - p}{0.02} \right) = \frac{-1}{0.02} = -50
\]

### Step 3: Plug into the Elasticity Formula
Now, substitute \( \frac{dx}{dp} = -50 \) into the elasticity formula:

\[
E(p) = \left( \frac{p}{x} \right) \cdot (-50)
\]

### Step 4: Express \( x \) in terms of \( p \)
From the previous expression for \( x \):

\[
x = \frac{\sqrt{6} - p}{0.02}
\]

### Step 5: Final Expression for Elasticity
Now substitute \( x \) into the elasticity formula:

\[
E(p) = \left( \frac{p}{\frac{\sqrt{6} - p}{0.02}} \right) \cdot (-50)
\]

Simplify:

\[
E(p) = \left( \frac{0.02p}{\sqrt{6} - p} \right) \cdot (-50)
\]

\[
E(p) = \frac{-50 \cdot 0.02p}{\sqrt{6} - p}
\]

\[
E(p) = \frac{-p}{\frac{\sqrt{6} - p}{50}}
\]

Thus, the elasticity of demand \( E(p) \) is:

\[
E(p) = \frac{-p}{\sqrt{6} - p}
\]

---

Would you like me to provide further explanation or details? Here are some related questions you might find helpful:

1. What is the interpretation of the elasticity of demand in this context?
2. How does the value of \( E(p) \) determine whether the demand is elastic or inelastic?
3. What happens to elasticity as the price increases?
4. How can we interpret the demand function geometrically?
5. What is the significance of elasticity in setting prices for a product?

**Tip**: Elasticity greater than 1 means the product is elastic, while elasticity less than 1 indicates inelastic demand.

The demand function for a certain make of exercise bicycle sold exclusively through cable television is p = sqrt(6) − 0.02x (0 ≤ x ≤ 300) where p is the unit price in hundreds of dollars and x is the quantity demanded/week. Compute the elasticity of demand E(p).

Learn how to compute the elasticity of demand for an exercise bicycle using the demand function p = sqrt(6) − 0.02x. This step-by-step solution covers key concepts in algebra and calculus, including differentiation and the formula for elasticity. Suitable for college-level economics or advanced high school calculus students.

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