Final answer:
The question involves calculating the price elasticity of demand for a given demand function at a price of 40. This requires differentiation of the demand function, substitution of the price into that derivative and the original function, and then using those values in the elasticity formula.
Step-by-step explanation:
The student is asking to evaluate the price elasticity of demand for an item at a specific price (p=40) where the demand function is given as Q=100e−0.02p. Price elasticity of demand (E) is represented by the formula E=(dp/dQ)×(Q/p). To find the elasticity at p=40, we first differentiate the demand function with respect to price 'p' to find dp/dQ. Then we substitute p=40 into both the derivative and the original demand function to get dp/dQ and Q, respectively, and plug these values into the elasticity formula.
Here is the process broken down step-by-step:
- First, find the derivative of Q with respect to 'p'.
- Substitute p=40 into the derivative to get dp/dQ.
- Calculate the demand Q at p=40.
- Finally, use the elasticity formula, substituting in the values for dp/dQ, Q, and p=40.
The result will tell us the elasticity at p=40, which indicates how sensitive the quantity demanded is to price changes at that price level. If the elasticity is less than one, the demand curve is inelastic in this area, meaning that the percentage change in quantity demanded is less than the percentage change in price.