Answer:
To find the price elasticity of demand at a specific point, we need to use the formula:
E = (dq/dp) * (p/q)
Where:
E = Price elasticity of demand
dq/dp = Derivative of the quantity demanded with respect to price
p = Price
q = Quantity demanded
Let's proceed with finding the price elasticity of demand at q = 9 for the given demand function p = 500 - 4q^2.
First, we need to find dq/dp by taking the derivative of q with respect to p:
dq/dp = -8q
Substituting q = 9 into dq/dp:
dq/dp = -8(9) = -72
Next, we calculate p/q by substituting q = 9 into the demand function:
p/q = (500 - 4(9)^2) / 9 = (500 - 4(81)) / 9 = (500 - 324) / 9 = 176/9
Now, we can find the price elasticity of demand at q = 9:
E = (dq/dp) * (p/q) = (-72) * (176/9) = -1408/3 ≈ -469.33
Since the price elasticity of demand is a negative value, we disregard the negative sign to determine the elasticity type. In this case, the price elasticity of demand at q = 9 is approximately 469.33.
Step-by-step explanation:
Interpreting the result:
- If the absolute value of E is less than 1 (|E| < 1), demand is inelastic. This means that a change in price will have a relatively smaller percentage change in quantity demanded.
- If the absolute value of E is equal to 1 (|E| = 1), demand is unit elastic. This means that a change in price will have an equal percentage change in quantity demanded.
- If the absolute value of E is greater than 1 (|E| > 1), demand is elastic. This means that a change in price will have a relatively larger percentage change in quantity demanded.
In our case, as the calculated price elasticity of demand at q = 9 is approximately 469.33, which is greater than 1, we can conclude that the demand is elastic. This implies that a change in price will lead to a relatively larger change in quantity demanded.