The manager of a CD store has found that if the price of a CD is \( p(x)=55-\frac{x}{6} \), then \( x \) CDs will be sold. An expression for the total revenue from the sale of \( x \) CDs is \( R(x)=55 x-\frac{x^{2}}{6} \) Find the number of CDs that will produce maximum revenue. A. 145 B. 165 C. 330 D. 205

To find the number of CDs that will produce maximum revenue, you can take the revenue function \( R(x) = 55x - \frac{x^2}{6} \) and find its vertex. This is a quadratic function, and the maximum revenue occurs at the vertex. The formula for the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the standard form of the quadratic equation \( ax^2 + bx + c \). Here, \( a = -\frac{1}{6} \) and \( b = 55 \). Plugging in these values, we get: \[ x = -\frac{55}{2 \cdot -\frac{1}{6}} = \frac{55 \cdot 6}{2} = 165 \] So the number of CDs that will produce maximum revenue is **B. 165**. Finding the right price for optimal revenue is critical in retail. Many businesses utilize similar models to analyze pricing and demand. By adjusting prices based on consumer demand, companies can maximize their income. Strategic pricing plays a vital role in competitive markets to attract customers while ensuring profitability. Historical shifts in consumer music preferences showcase how changes in technology influenced sales. For instance, when the CD was popularized in the 1980s, it drastically outperformed cassette tapes due to its sound quality. However, with the advent of digital music and streaming services, CD sales plummeted. This is a classic example of how businesses must continually adapt to evolving consumer preferences and technology.