Problem 22 - Applied Calculus

Question

Annual sales of music compact discs (CDs) have declined since \(2000 .\) Sales were 942.5 million in 2000 and 384.7 million in \(2008 .^{17}\) (a) Find a formula for annual sales, \(S\), in millions of music CDs, as a linear function of the number of years, \(t,\) since 2000 (b) Give units for and interpret the slope and the vertical intercept of this function. (c) Use the formula to predict music CD sales in 2012

Step-by-Step Solution

Answer

The annual sales formula is \(S = -69.725t + 942.5\). By 2012, sales dropped to approximately 105.8 million CDs.

Step 1: Identify Known Values

First, identify the known values from the problem statement. In the year 2000, sales were 942.5 million, and in 2008, sales were 384.7 million. Let these years correspond to the values \((0, 942.5)\) and \((8, 384.7)\) for our calculation.

Step 2: Calculate Slope of the Line

To find the slope (rate of change) of the linear function, use the formula for the slope between two points, \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let 2000 be \(t = 0\) and 2008 be \(t = 8\):\[ m = \frac{384.7 - 942.5}{8 - 0} = \frac{-557.8}{8} = -69.725 \]

Step 3: Construct Linear Equation

Using the point-slope form, \(y - y_1 = m(x - x_1)\), and substituting one of the points (0, 942.5):\[ S - 942.5 = -69.725(t - 0) \]Simplify to obtain the linear equation:\[ S = -69.725t + 942.5 \]

Step 4: Interpret Slope and Intercept

The slope \(-69.725\) million CDs per year indicates that sales decreased by approximately 69.725 million CDs each year. The vertical intercept 942.5 million represents the initial sales in the year 2000 when \(t = 0\).

Step 5: Predict Sales in 2012

Substitute \(t = 12\) (since 2012 is 12 years after 2000) into the linear equation to find the sales:\[ S = -69.725(12) + 942.5 \]Calculate to find:\[ S = -836.7 + 942.5 = 105.8 \] million CDs.

Key Concepts

Slope Calculation

Understanding slope is crucial when dealing with linear functions. It tells us how quickly or slowly something changes over time. In this example, the slope describes the decline in music CD sales from 2000 to 2008. The formula to find the slope, which is usually denoted as \( m \), is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) represent two points on the line.

From the problem, we have the points \((0, 942.5)\) and \((8, 384.7)\). Substituting these into the slope formula gives \[ m = \frac{384.7 - 942.5}{8 - 0} = \frac{-557.8}{8} = -69.725 \]. This result means sales decreased by approximately 69.725 million CDs per year.

The negative sign indicates a decline. Interpreting slope in real-world terms helps us understand trends, such as decreasing sales in various industries. It provides a clear picture of the rate of change over time.

Interpretation of Intercept

Intercepts in linear equations offer valuable insights. They are points where lines intersect the axes. In our equation \( S = -69.725t + 942.5 \), the intercept is the constant term, 942.5. This number represents the point where the line crosses the vertical or \( y \)-axis when \( t = 0 \).

The vertical intercept of 942.5 million CDs tells us the sales at the starting point, the year 2000. This point reflects the initial sales before any changes occur due to the slope's influence over time. Interpreting this helps in establishing a baseline, from which we can understand how much sales have changed.

Intercepts are often crucial in predictive models as they provide a starting reference point. Understanding them aids in grounding expectations about future values by showing where the relationship depicted by the function begins.

Predictive Modelling

Predictive modelling uses known data to forecast future outcomes. In the context of our linear function for CD sales, predictive modelling allows us to estimate future sales. The linear equation \( S = -69.725t + 942.5 \) can be used to predict future CD sales by plugging in values of \( t \), the number of years since 2000.

For instance, to predict 2012 sales, set \( t = 12 \) (since 2012 is 12 years after 2000). Substituting into the equation gives \( S = -69.725(12) + 942.5 = 105.8 \) million CDs. This prediction aligns with the ongoing sales trend identified by the equation.

Predictive models are pivotal in planning and decision-making processes, offering insights based on past data. They are employed across various fields to forecast outcomes, providing guidance for future strategy. In this problem, it helps understand the anticipated trajectory of an industry over time.