Question: The price $p$ (in dollars) and the quantity $x$ sold of a certain product satisfy the demand equation $x = -6p + 300$. Answer parts (a) through (g).
Part 1 of 7
(a) Find a model that expresses the revenue $R$ as a function of $p$. (Remember, $R = xp$.)
$R(p) =$
1. **State the problem:** We are given the demand equation $x = -6p + 300$ where $p$ is the price and $x$ is the quantity sold. We need to find the revenue function $R(p)$, where revenue $R$ is the product of price and quantity sold, i.e., $R = xp$.
2. **Write the formula for revenue:**
$$R = x imes p$$
3. **Substitute the demand equation into the revenue formula:**
$$R(p) = (-6p + 300) imes p$$
4. **Distribute $p$ across the terms:**
$$R(p) = -6p^2 + 300p$$
5. **Simplify the expression:** The expression is already simplified with integer coefficients.
6. **Final revenue function:**
$$\boxed{R(p) = -6p^2 + 300p}$$
This function expresses revenue as a quadratic function of price $p$.