To find the price elasticity of demand at a specific point using the point slope method, we first need to find the quantity demanded at the given price, calculate the derivative of the demand function, and then apply the elasticity formula.
Given the demand equation: \[ QD = 10Q - 4P \]
We need to rearrange this equation in terms of \( P \).
- Rearranging the equation: \[ 4P = 10Q - QD \] \[ P = \frac{10Q - QD}{4} \]
Next, we'll assume \( QD \) is equal to \( Q \) when we calculate elasticity, and we need to solve for \( Q \) when \( P = 10 \) (price given):
- Substituting \( P = 10 \): \[ 10 = \frac{10Q - QD}{4} \] Multiply both sides by 4: \[ 40 = 10Q - QD \] Thus, rearranging gives us: \[ QD = 10Q - 40 \]
Now, we need to find the value of \( Q \) such that \( QD = Q \): \[ Q = 10Q - 40 \] \[ 40 = 9Q \] \[ Q = \frac{40}{9} \approx 4.44 \]
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Finding the derivative \( \frac{dQD}{dP} \): To calculate price elasticity of demand, we also require \( \frac{dQD}{dP} \), which is the slope of the demand curve. From the equation \( QD = 10Q - 4P \), we isolate P: \[ QD + 4P = 10Q \] Differentiate with respect to P: Taking the derivative of both sides: \[ \frac{dQD}{dP} = -4 \]
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Price Elasticity of Demand: We can use the formula for price elasticity of demand: \[ E_d = \frac{dQD}{dP} \cdot \frac{P}{QD} \] Substituting the values we've found:
- \( \frac{dQD}{dP} = -4 \)
- \( P = 10 \)
- \( QD = Q \approx 4.44 \)
Thus: \[ E_d = -4 \cdot \frac{10}{4.44} \] Calculating that, \[ E_d \approx -4 \cdot 2.25 \approx -9 \]
Therefore, the price elasticity of demand at the price of $10 is approximately \(-9\), indicating that the demand is highly elastic.
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