Questions: Use the price-demand equation (p+0.05 x=130, 0 leq p leq 130). Find all values of (p) for which demand is elastic. The demand is elastic on (Type your answer in interval notation.)

Solution Steps

From the price-demand equation \( p + 0.05x = 130 \), we can express the quantity demanded \( x \) in terms of price \( p \):

\[ x = 2600 - 20p \]

Next, we calculate the derivative of \( x \) with respect to \( p \):

\[ \frac{dx}{dp} = -20 \]

Using the price elasticity of demand formula:

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

Substituting \( x \) and \( \frac{dx}{dp} \):

\[ E(p) = \frac{p}{2600 - 20p} \cdot (-20) = -\frac{20p}{2600 - 20p} \]

To find where demand is elastic, we solve the inequality:

\[ |E(p)| > 1 \]

This leads to:

\[ 20 \cdot \left| \frac{p}{2600 - 20p} \right| > 1 \]

This simplifies to:

\[ \frac{p}{2600 - 20p} > \frac{1}{20} \quad \text{or} \quad \frac{p}{2600 - 20p} < -\frac{1}{20} \]

Solving the first inequality gives:

\[ p > 65 \quad \text{and} \quad p \neq 130 \]

The second inequality does not yield any valid solutions within the given constraints.

Final Answer