Questions: If P(x)=4+x-4 x^2 represents the profit in selling x thousand Boombotix speakers, how many speakers should be sold to maximize profit?

Transcript text: If $P(x)=4+x-4 x^{2}$ represents the profit in selling $x$ thousand Boombotix speakers, how many speakers should be sold to maximize profit? $\square$

Solution

Solution Steps

To maximize the profit represented by the quadratic function $ P(x) = 4 + x - 4x^2 $, we need to find the vertex of the parabola. Since the coefficient of $ x^2 $ is negative, the parabola opens downwards, and the vertex will give the maximum point. The x-coordinate of the vertex for a quadratic equation $ ax^2 + bx + c $ is given by $ x = -\frac{b}{2a} $.

Step 1: Identify the Profit Function

The profit function is given by $ P(x) = 4 + x - 4x^2 $, where $ x $ represents the number of thousands of Boombotix speakers sold.

Step 2: Determine the Vertex of the Parabola

To find the number of speakers that maximizes profit, we need to find the vertex of the parabola. The x-coordinate of the vertex for a quadratic function $ ax^2 + bx + c $ is given by:

\[ x = -\frac{b}{2a} \]

For the given function, $ a = -4 $ and $ b = 1 $. Substituting these values, we get:

\[ x = -\frac{1}{2 \times (-4)} = \frac{1}{8} = 0.125 \]

Step 3: Convert to Number of Speakers

Since $ x $ represents thousands of speakers, the actual number of speakers is:

\[ 0.125 \times 1000 = 125 \]