To find the Elasticity of Demand at a price of \[tex]$41, we need to follow these steps:
1. Identify the demand function:
\[
D(p) = \sqrt{100 - 2p}
\]
2. Find the derivative of the demand function with respect to price \( p \):
\[
\frac{dD}{dp} = \frac{d}{dp} \left( \sqrt{100 - 2p} \right)
\]
Using the chain rule:
\[
\frac{dD}{dp} = \frac{1}{2} \left( 100 - 2p \right)^{-\frac{1}{2}} \cdot (-2) = -\frac{1}{\sqrt{100 - 2p}}
\]
3. Plug in the price \( p = 41 \) into the demand function to find the quantity demanded at this price:
\[
D(41) = \sqrt{100 - 2 \cdot 41} = \sqrt{100 - 82} = \sqrt{18}
\]
4. Calculate the derivative at price \( p = 41 \):
\[
\frac{dD}{dp}\Bigg|_{p=41} = -\frac{1}{\sqrt{100 - 2 \cdot 41}} = -\frac{1}{\sqrt{18}}
\]
5. Determine the elasticity of demand \( E \) using the formula:
\[
E = \left( \frac{p}{D(p)} \right) \cdot \frac{dD}{dp}
\]
Plug in the values we know:
\[
E = \left( \frac{41}{\sqrt{18}} \right) \cdot \left( -\frac{1}{\sqrt{18}} \right)
\]
\[
E = 41 \cdot \left( \frac{-1}{18} \right)
\]
\[
E \approx -2.277
\]
6. Interpret the elasticity of demand:
- If \( E > 1 \), the demand is elastic.
- If \( E = 1 \), the demand is unitary.
- If \( E < 1 \), the demand is inelastic.
Given \( E \approx -2.277 \), the absolute value \( |E| = 2.277 \), which is greater than 1.
7. Determine the nature of the demand:
Since \( |E| > 1 \), the demand at the price of \$[/tex]41 is inelastic.
8. Decision to increase revenue:
- For inelastic demand ([tex]\( |E| < 1 \)[/tex]), raising prices increases revenue.
- For elastic demand ([tex]\( |E| > 1 \)[/tex]), lowering prices increases revenue.
- For unitary demand ([tex]\( |E| = 1 \)[/tex]), changing prices does not affect revenue.
Given that the demand is inelastic, to increase revenue, we should Raise Prices.
### Summary:
1. The elasticity of demand at a price of \$41 is approximately -2.277.
2. At this price, the demand is inelastic.
3. Based on this, to increase revenue, we should Raise Prices.
