Problem 93 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The sales (in billions of dollars) in restaurants and bars in California from the beginning of \(1993(t=0)\) through \(2000(t=7)\) are approximated by the function $$ S(t)=0.195 t^{2}+0.32 t+23.7 \quad(0 \leq t \leq 7) $$ a. Show that the sales in restaurants and bars continued to rise after smoking bans were implemented in restaurants in 1995 and in bars in 1998 . Hint: Show that \(S\) is increasing in the interval \((2,7)\). b. What can you say about the rate at which the sales were rising after smoking bans were implemented?

Step-by-Step Solution

Answer

The sales in restaurants and bars continued to rise after smoking bans were implemented as the derivative of the function \(S(t) = 0.195t^2 + 0.32t + 23.7\) is positive in the interval \(2 \leq t \leq 7\) (\(\frac{dS(t)}{dt} = 0.39t + 0.32\)). The rate at which the sales were rising was also increasing because the derivative is a linear function of \(t\). Therefore, the sales were rising at an increasingly faster pace over the years.

Step 1: Find the derivative of \(S(t)\)

To check whether the function is increasing in the given interval, we need to find its derivative. Differentiate \(S(t)\) with respect to \(t\): \[ \frac{dS(t)}{dt} = \frac{d(0.195t^2 + 0.32t + 23.7)}{dt}. \] ##Step 2: Calculate the derivative##

Step 2: Calculate the first derivative of \(S(t)\)

Using the power rule and the linearity of differentiation, we find the first derivative of \(S(t)\): \[ \frac{dS(t)}{dt} = 0.39t + 0.32. \] We want to check if this derivative is positive throughout the interval \(2 \leq t \leq 7\). ##Step 3: Check the sign of the derivative in the interval##

Step 3: Check the sign of \(\frac{dS(t)}{dt}\) in the interval \(2 \leq t \leq 7\)

Plug in the minimum value of the interval, \(t=2\), into the derivative: \[ \frac{dS(t)}{dt} \bigg|_{t = 2} = 0.39(2) + 0.32 = 1.1. \] Since the first derivative is positive in 1995, the sales are increasing after the smoking ban in restaurants. Now, plug in the other critical year, \(t=5\), into the derivative (1998 corresponds to \(t=5\)): \[ \frac{dS(t)}{dt} \bigg|_{t = 5} = 0.39(5) + 0.32 = 2.17. \] Since the first derivative is positive in 1998, the sales are increasing after the smoking ban in bars. The derivative is also positive for all other values in the interval \(2 \leq t \leq 7\). Hence, we conclude that the sales in restaurants and bars continued to rise after smoking bans were implemented. #b. What can you say about the rate at which the sales were rising?#

Step 4: Interpret the derivative of \(S(t)\)

Since the derivative of \(S(t)\) is positive in the interval \(2 \leq t \leq 7\), we can say that the sales in restaurants and bars were rising throughout this period. And since \(\frac{dS(t)}{dt}\) is a linear function of \(t\), we can say that the rate at which the sales were rising was also increasing. In other words, not only were the sales increasing after smoking bans were implemented, but they were rising at an increasingly faster pace over the years.

Key Concepts

Derivatives

Derivatives are a fundamental concept in calculus, serving as a tool to measure the rate of change of a function with respect to a variable. Simply put, if you have a function, the derivative shows how the function's output changes as you wiggle its input a little bit. Here’s how derivatives are typically used:

• To determine if a function is increasing or decreasing: A positive derivative means the function is increasing, while a negative one means it's decreasing.
• In optimization problems: Derivatives help find maximum or minimum values of functions.
• For curve sketching: They provide information about the slope of the tangent line to the graph of the function, helping to understand the shape of the graph.

To calculate a derivative, we often apply differentiation rules. One common rule is the power rule, which states that the derivative of a term like \( ax^n \) is \( nax^{n-1} \). Understanding derivatives allows us to gain insight into how things change over time or other parameters.

Function Analysis

Function analysis involves examining the properties and behaviors of mathematical functions. Through this, we gain insights into how inputs are transformed into outputs. When analyzing functions:

• We often determine if outputs are increasing or decreasing over certain intervals by examining the sign of the derivative.
• We check for critical points where the derivative is zero. These points can indicate potential maximums, minimums, or inflection points.
• We explore the end behavior to understand how the function behaves as inputs grow large or small. This can be particularly important for anticipating trends.

By analyzing functions, especially polynomial ones like in our example \( S(t) = 0.195t^2 + 0.32t + 23.7 \), we observe how it models real-world scenarios, showing trends within specific time intervals. In this case, the derivative's positivity informs us that sales steadily increase in the given timeline.

Economic Trends

Economic trends refer to patterns or movements observed in economic data over a period of time. They help in understanding economic activities, like sales, which can be influenced by numerous factors such as policy changes or consumer behaviors. Here are key points where calculus and derivatives play their role in assessing trends:

• Derivatives assess changing rate of sales over time, guiding business strategy.
• Increasing linear derivative values illustrate accelerating growth in certain periods, pointing to strengthening trends.
• Such analysis helps economists or business managers in forecasting by identifying growth phases or potential slowdowns.

In the context of the problem given, even after smoking bans, the derivative indicates increased sales, suggesting that these bans did not negatively impact the economic trend of rising sales in restaurants and bars.

Smoking Bans Impact

The implementation of smoking bans has been a subject of analysis to determine their impact on industries such as restaurants and bars. Despite initial concerns, data analysis often shows that bans may not deter growth. Key considerations include:

• Public health improvements due to reduced involuntary exposure to secondhand smoke, potentially bringing more non-smoking patrons.
• Long-term economic benefits can emerge from healthier populations that demand more diversified services and experiences.
• Increased sales post-ban, as shown by positive derivative values over time, indicate resilience and adaptation of businesses to new regulations.

Understanding how such regulations intersect with industry trends can provide valuable insights into regulatory impacts on health and economics. Comprehensive analysis is crucial to determine whether bans have positive, neutral, or negative financial implications.