Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Motorola The sales per share \(S\) (in dollars) for Motorola from 1992 to 2005 can be approximated by the function \(S=\left\\{\begin{array}{lr}2.33-0.909 t+10.394 \ln t, & 2 \leq t \leq 10 \\ 0.6157 t^{2}-15.597 t+110.25, & 11 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=2\) corresponding to 1992\. (Source: Motorola) (a) Use a graphing utility to graph the function. (b) Describe the change in sales per share that occurred in 2001 .

Step-by-Step Solution

Answer

Without precise function graph and numeric calculations, can't provide the exact short answer. However, the short answer would be the quantified change in sales per share from 2000 to 2001 obtained in Step 3, along with an interpretation based on the graph and overall trend observed.

Step 1: Understand the given function

Firstly, need to understand that the function \(S\) that approximates the sales per share is piecewise, which means it operates differently in different intervals of \(t\). From \(t=2\) to \(t=10\) (which corresponds to years 1992 to 2000), the function is \(S=2.33-0.909t+10.394 \ln t\). From \(t=11\) to \(t=15\) (which corresponds to years 2001 to 2005), the function is \(S=0.6157 t^{2}-15.597 t+110.25\).

Step 2: Graph the function using a graphing utility

Next, to graph these functions, substitute values of \(t\) within the defined intervals into the given functions using a graphing tool. Observe how the function behaves within the respective intervals and compare the function's progression from the period 1992-2000 to the period 2001-2005.

Step 3: Determine the sales per share in 2001

In order to describe the change in sales in 2001, calculate the value of \(S\) at \(t=11\) (2001), which falls in the second interval of \(t\). Substitute \(t=11\) in \(S=0.6157 t^{2}-15.597 t+110.25\), and compare this value to the value of \(S\) at \(t=10\) (2000), which would be obtained from the first function \(S=2.33-0.909t+10.394 \ln t\). The difference between these two values will illustrate the change in sales per share from 2000 to 2001.

Step 4: Interpret the graph and the change in sales per share

Based on the graph and the computed change in the sales per share, provide a description of how the sales per share changed in 2001 in the context of the company's overall sales trend from 1992-2005.

Key Concepts

Using Graphing Utilities Effectively

Graphing utilities are essential tools for visually understanding mathematical functions. These tools allow you to plot complex equations and witness their behavior over specified intervals. For this exercise, we deal with a piecewise function, which behaves differently in two specific intervals of time. To graph a piecewise function, input each part separately into the graphing utility, ensuring you specify the correct interval for each segment.

When entering the first part of this function, which applies from 1992 to 2000 (\(t=2\) to \(t=10\)), use the equation \(S=2.33-0.909t+10.394 \ln t\). For the second interval, from 2001 to 2005 (\(t=11\) to \(t=15\)), use \(S=0.6157 t^{2}-15.597 t+110.25\).

The graphing utility will help you see transitions between periods. It visualizes how shifts from one part of the function to the next occur. Observe any sharp changes, trends, or significant differences in the slope between intervals. This graphical representation aids in better understanding the behavior of sales per share over time.

Understanding Sales Per Share

Sales per share (SPS) is a vital metric in evaluating a company's financial health and performance over time. It represents how much sales are generated per share of stock, offering insights into the efficiency and productivity of a company's operations.

In the context of this problem, Motorola's sales per share are examined throughout a period between 1992 and 2005. It allows students to see how the company's sales evolved over these years, especially around significant points like the year 2001. Note that calculating SPS involves using the given functions for specific time periods and comparing these figures at different points.

• The first function, valid from 1992 to 2000, is based on a natural logarithm, indicating some level of exponential relationship within that time.
• The second from 2001 to 2005 is a quadratic equation reflecting different growth dynamics, possibly due to shifting market conditions or internal company changes.

Tracking these changes between consecutive years, particularly 2000 to 2001, gives insight into the fluctuations in Motorola's sales per share, highlighting any dramatic increase or decrease.

Grasping Exponential Functions

Exponential functions are mathematical expressions where variables appear in exponent positions. They define how quantities grow or decay at consistent rates, often used in modeling compound growth scenarios.

In the Motorola problem, part of the function \(S=2.33-0.909t+10.394 \ln t\) indirectly incorporates an exponential element through the natural log \(\ln\) function. This suggests that sales per share are subject to exponential changes during the first interval (1992-2000).

Why is this important? Because exponential growth reflects real-world scenarios where initial increases can be slow, but soon transform into rapid growth. Conversely, exponential decay might showcase a fast decline initially, followed by stabilization. As a student, recognizing these patterns assists in predicting and interpreting future trends based on historical data, especially in financial contexts like sales projections.