Problem 203 - Calculus Volume 1

Question

The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by \(y=10+5 \sin x\) where \(y\) is the number of hamburgers sold and \(x\) represents the number of hours after the restaurant opened at 11 a.m. until 11 \(\mathrm{p.m.}\) , when the store closes. Find \(y^{\prime}\) and determine the intervals where the number of burgers being sold is increasing.

Step-by-Step Solution

Answer

The number of burgers sold is increasing when 0 < x < 1.57 and 4.71 < x < 6.28.

Step 1: Differentiate the function

Given the function \(y = 10 + 5 \sin x\). To find \(y'\), differentiate \(y\) with respect to \(x\). The derivative of a constant is zero, and the derivative of \(\sin x\) is \(\cos x\). Thus, \(y' = 5 \cos x\).

Step 2: Identify the increasing intervals

A function is increasing where its derivative is positive. Thus, \(y\) is increasing where \(y' = 5 \cos x > 0\). This means \(\cos x > 0\).

Step 3: Establish the interval domain

Since the restaurant is open from 11 a.m. to 11 p.m., \(x\) ranges from 0 to 12 hours (0 corresponds to 11 a.m. and 12 corresponds to 11 p.m.).

Step 4: Solve for intervals of positivity

\(\cos x\) is positive when \(x\) lies between \(2n\pi\) and \(((2n + 1)\pi)\) for integers \(n\). Given the range \(0 \leq x \leq 12\), these intervals within one period \([0, 2\pi]\) correspond to \((0, \frac{\pi}{2})\) and \((\frac{3\pi}{2}, 2\pi)\). Convert these intervals using \(x/\pi \approx 3.818\), resulting in 0 to approximately 3.818 hours and 9.545 to 12 hours.

Step 5: Define numerical intervals

The interval \((0, \frac{\pi}{2})\) converts to approximately \(x \in (0, 1.57)\). Similarly, \((\frac{3\pi}{2}, 2\pi)\) converts to \(x \in (4.71, 6.28)\). Thus, the number of burgers sold is increasing between 0 to approximately 1.57 hours after opening, and 4.71 to 6.28 hours after opening, assuming the opening time is 11 a.m.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that deals with finding the rate at which a function is changing at any given point. This rate of change is known as the derivative. In the exercise, the function representing the number of hamburgers sold is given by:

• \( y = 10 + 5 \sin x \)

To differentiate this function with respect to \( x \), we need to apply basic differentiation rules. The derivative of a constant, like 10, is zero because constants do not change. For the term \( 5 \sin x \), we utilize the fact that the derivative of \( \sin x \) is \( \cos x \). Thus the overall derivative \( y' \) becomes:

• \( y' = 5 \cos x \)

Understanding differentiation allows us to determine how the hamburger sales change over time. Specifically, the derivative \( y' = 5 \cos x \) gives us a way to analyze when sales are increasing or decreasing.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are essential in analyzing periodic phenomena that occur in cycles. In the given exercise, the function \( 5 \sin x \) reflects the periodic nature of hamburger sales. Sine and cosine functions are related, and knowing their derivatives and properties is crucial in calculus.The sine function reaches its peak and trough in a pattern, and the cosine function, which is the derivative of the sine function, helps us evaluate these changes. When calculating \( y' = 5 \cos x \), we emphasize that the sine and cosine functions have specific intervals where they are positive or negative. The cosine function, in particular, cycles between -1 and 1, and the value of \( \cos x \) determines when the sales are increasing or decreasing:

• \( \cos x > 0 \) indicates a period of increasing number of sales
• \( \cos x < 0 \) points to a declining number of sales

This cyclical behavior is what makes trigonometric functions so valuable in mathematical analysis and their applications in real-world scenarios, such as the one described.

Interval Analysis

Interval analysis focuses on finding specific ranges where a certain condition holds. In terms of calculus, we analyze intervals to determine where a function is increasing or decreasing. Using the derivative \( y' = 5 \cos x \), we need to identify when this derivative is positive.Finding the intervals involves the function of cosine. We know that \( \cos x \) is positive between 0 and \( \frac{\pi}{2} \) and between \( \frac{3\pi}{2} \) and \( 2\pi \) within each cycle of \( 0 \leq x < 2\pi \). Since the restaurant operates between 11 a.m. and 11 p.m., the interval for \( x \) is from 0 to 12.Translating from radians to hours, these intervals are 0 to about 1.57 hours (related to \( \frac{\pi}{2} \)) and 4.71 to about 6.28 hours (correlating to \( \frac{3\pi}{2} \)) within a day. Therefore, sales are increasing during these times:

• From opening to approximately 1.57 hours later
• And again from roughly 4.71 hours to 6.28 hours

By understanding these intervals, one can predict and understand the patterns in sales dynamics.