Final answer:
To express q as a function of p, rearrange the demand equation and isolate q. Calculate the price elasticity of demand using the given recipe and interpret the result. Determine the changes in price needed to increase revenue based on the elasticity value.
Step-by-step explanation:
To express q as a function of p, we can rearrange the given demand equation: p = 41 / q¹.⁵. By isolating q on one side of the equation, we get q = (41 / p)²/³. Therefore, q is a function of p represented by q = (41 / p)²/³.
The price elasticity of demand can be calculated using the formula: E = (dq / dp) * (p / q). Substituting the given price p = $3.70 into the equation, we can calculate the elasticity E as follows:
- Calculate dq / dp by taking the derivative of q with respect to p: dq / dp = (-82 / 225) * (41 / p)²/³.
- Substitute the values of dp = -0.30 (a 30 cents decrease in price) and p = $3.70 into the derivative equation to calculate dq / dp.
- Finally, substitute the values of p = $3.70, q, and dq / dp into the elasticity formula to calculate the price elasticity of demand at the given price.
The interpretation of the calculated elasticity value will provide information on how demand for the product responds to changes in price. If the elasticity is greater than 1, it means the product has elastic demand, indicating that a price decrease could lead to an increase in revenue.
If the elasticity is less than 1, the product has inelastic demand, and a price increase could lead to an increase in revenue. If the elasticity is equal to 1, the demand is unitary elastic, and the revenue is already maximized at the current price.