Questions: The price-demand function for the sale of yo-yos is: p=5-0.01x where p is the price of a yo-yo in dollars, and x is the demand for yo-yos at a price of p dollars. a. R'(280)= b. What are the correct units for R'(280)? Select an answer

Solution Steps

To solve the given problem, we need to follow these steps:

a. To find \( R'(280) \), we first need to determine the revenue function \( R(x) \). The revenue function is given by \( R(x) = p \cdot x \). We then differentiate \( R(x) \) with respect to \( x \) to find \( R'(x) \) and evaluate it at \( x = 280 \).

b. The units for \( R'(280) \) can be determined by understanding the units of the revenue function and its derivative.

1. Determine the revenue function \( R(x) \) using the given price-demand function.
2. Differentiate \( R(x) \) with respect to \( x \) to find \( R'(x) \).
3. Evaluate \( R'(x) \) at \( x = 280 \).
4. Identify the units of \( R'(x) \).

The price-demand function is given by: \[ p = 5 - 0.01x \] The revenue function \( R(x) \) is defined as: \[ R(x) = p \cdot x = (5 - 0.01x) \cdot x = 5x - 0.01x^2 \]

To find the rate of change of revenue with respect to demand, we differentiate \( R(x) \): \[ R'(x) = \frac{d}{dx}(5x - 0.01x^2) = 5 - 0.02x \]

Now, we evaluate \( R'(x) \) at \( x = 280 \): \[ R'(280) = 5 - 0.02 \cdot 280 = 5 - 5.6 = -0.6 \]

The units of \( R'(280) \) represent the change in revenue per unit change in demand, which is expressed in dollars per yo-yo.

Final Answer