Given the following demand functions, calculate the price elasticity of demand when P = 10,
Qd = 100 − 5P
Qd = 50P − P
0.5

To calculate the price elasticity of demand (PED), we first need to use the formula:

\[
E_d = \frac{dQ_d}{dP} \times \frac{P}{Q_d}
\]

Where:
- \(E_d\) is the price elasticity of demand,
- \(\frac{dQ_d}{dP}\) is the derivative of the demand function with respect to price (\(P\)),
- \(P\) is the price,
- \(Q_d\) is the quantity demanded at price \(P\).

We will do this for both given demand functions, one at a time.

### 1. Demand Function 1: \(Q_d = 100 - 5P\)

1. Find \(\frac{dQ_d}{dP}\):
   \[
   \frac{dQ_d}{dP} = -5
   \]

2. Substitute \(P = 10\) into the demand function to find \(Q_d\):
   \[
   Q_d = 100 - 5(10) = 100 - 50 = 50
   \]

3. Apply the elasticity formula:
   \[
   E_d = -5 \times \frac{10}{50} = -5 \times 0.2 = -1
   \]

Thus, the price elasticity of demand for the first function is \(-1\).

### 2. Demand Function 2: \(Q_d = 50P - P^{0.5}\)

1. Find \(\frac{dQ_d}{dP}\):
   \[
   \frac{dQ_d}{dP} = 50 - \frac{1}{2}P^{-0.5}
   \]

2. Substitute \(P = 10\) into the demand function to find \(Q_d\):
   \[
   Q_d = 50(10) - (10)^{0.5} = 500 - 3.162 \approx 496.838
   \]

3. Substitute \(P = 10\) into \(\frac{dQ_d}{dP}\):
   \[
   \frac{dQ_d}{dP} = 50 - \frac{1}{2}(10^{-0.5}) = 50 - \frac{1}{2}(0.3162) = 50 - 0.1581 \approx 49.8419
   \]

4. Apply the elasticity formula:
   \[
   E_d = 49.8419 \times \frac{10}{496.838} \approx 49.8419 \times 0.0201 \approx 1.002
   \]

Thus, the price elasticity of demand for the second function is approximately \(1.002\).

### Summary of Results:
- **Demand function 1:** Price elasticity of demand = \(-1\)
- **Demand function 2:** Price elasticity of demand = \(1.002\)

Would you like further clarification or more details on any step? 

Here are 5 related questions for further practice:
1. How would the elasticity change if the price increased to \(P = 15\) in the first function?
2. What does it mean if the price elasticity of demand is exactly \(-1\)?
3. Can you calculate the revenue-maximizing price from these demand functions?
4. How would the elasticity of the second function change if we altered the exponent on \(P^{0.5}\)?
5. How do we interpret a price elasticity of demand greater than 1?

**Tip:** Elasticities less than \(-1\) (in absolute value) imply that demand is elastic, meaning a small change in price leads to a large change in quantity demanded.

Given the following demand functions, calculate the price elasticity of demand when P = 10, Qd = 100 − 5P, Qd = 50P − P^0.5

Calculate the price elasticity of demand for two different demand functions at P = 10. Learn how to use the elasticity formula and derivatives to analyze demand sensitivity, covering both linear and non-linear functions. Suitable for college-level economics or advanced high school students.

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