Final answer:
The elasticity of demand at a price of $2 for the function D(p)=125-4p² is approximately -0.294, indicating that the demand at this price is inelastic.
Step-by-step explanation:
To find the elasticity of demand at a price of $2 for the demand function D(p)=125−4p², we first need to calculate the derivative of the demand function with respect to price (p), which represents the rate of change of quantity demanded in response to a change in price. The formula for elasticity of demand is given by:
E(p) = (p / Q(p)) * (dQ(p)/dp)
Firstly, calculate the derivative of the demand function:
dD(p)/dp = d(125 − 4p²)/dp = −8p
Now, substitute p = 2 into D(p) to find quantity demanded (Q):
Q(2) = 125 − 4(2)² = 125 − 4(4) = 125 − 16 = 109
Next, compute the derivative at p = 2:
dD(p)/dp|p=2 = −8(2) = −16
Finally, we use the elasticity formula to find the elasticity of demand at p = 2:
E(2) = (2 / 109) * (−16) = −32/109 ≈ −0.294
Since the value is less than 1, the demand is inelastic at this price level. This means that the quantity demanded does not change significantly with a change in price.