Revenue Maximizing Quantity Calculation

To find the revenue maximizing quantity sold, we need to find the value of $x$ that maximizes the revenue function $R$. The revenue function is a quadratic equation:

$R(x) = -75x^2 + 2250x + 300000$

This is a downward-opening parabola (since the coefficient of $x^2$ is negative, $-75 < 0$), so its maximum value occurs at the vertex.

Finding the Vertex

The x-coordinate of the vertex of a quadratic function in the form $ax^2 + bx + c$ is given by the formula:

$x = -\frac{b}{2a}$

In this revenue function:

• $a = -75$
• $b = 2250$
• $c = 300000$

Calculating Maximum Quantity

Substitute the values of $a$ and $b$ into the vertex formula:

$x = -\frac{2250}{2 \times (-75)}$ $x = -\frac{2250}{-150}$ $x = \frac{2250}{150}$ $x = 15$

The revenue maximizing level of quantity sold is 15 units.