Problem 53 - Precalculus: Functions and Graphs

Question

Graph the equation \(y=f(t)\) on the inter val \([0,24] .\) Let \(y\) represent the outdoor temperature \(\left(\text { in }^{\circ} \mathrm{F}\right)\) at time \(t\) (in hours), where \(t=0\) corresponds to 9 A.M. Describe the temperature during the 24 -hour interval. \(y=20+15 \sin \frac{\pi}{12} t\)

Step-by-Step Solution

Answer

The temperature fluctuates between 5°F and 35°F in a 24-hour period, peaking at 3 P.M. and dipping at 3 A.M.

Step 1: Identify the components of the equation

The given equation is \( y = 20 + 15 \sin \frac{\pi}{12} t \). It represents a sinusoidal function where 20 is the midline or equilibrium position of the sine curve, 15 is the amplitude, and \( \frac{\pi}{12} \) is the frequency component.

Step 2: Determine the amplitude and midline

The amplitude of the function is 15, which means the temperature fluctuates 15 units above and below the midline. The midline is 20, indicating that the average temperature over time is 20°F.

Step 3: Analyze the frequency and period

The frequency component \( \frac{\pi}{12} \) suggests that the sine completes one cycle every 24 hours. This is determined from the standard period formula for sine functions: \( \frac{2\pi}{\frac{\pi}{12}} = 24 \).

Step 4: Determine the behavior at specific times

At \( t = 0 \) (9 A.M.), \( y = 20+15 \sin(0) = 20 \). At \( t = 12 \) (9 P.M.), \( y = 20+15 \sin(\pi) = 20 \) again, indicating a return to the average. The maximum temperature occurs when \( \sin(\theta) = 1 \), which is at \( t = 6 \) and \( t = 18 \), giving a maximum of \( 35^{\circ} \text{F} \). The minimum temperature occurs at integer multiples of \( t = 12 \), producing a minimum of \( 5^{\circ} \text{F} \).

Step 5: Sketch the graph

The graph of this equation will be a sinusoidal wave, starting at 20°F at 9 A.M., reaching a maximum of 35°F at 3 P.M., returning to 20°F at 9 P.M., dipping to 5°F at 3 A.M., and returning to 20°F by the next 9 A.M. This reflects a cycle repeated daily.

Key Concepts

Sinusoidal Functions

The equation given describes a sinusoidal function. Sinusoidal functions are a type of periodic function that oscillate in a smooth, repetitive wave. The most well-known sinusoidal functions are sine and cosine. They are used to model phenomena that exhibit cyclic patterns, such as sound waves and tides.
In the equation \( y = 20 + 15 \sin \frac{\pi}{12} t \), the function is based on the sine curve. This means the temperature, represented by \( y \), will vary in a predictable, wave-like pattern over time \( t \). Such functions can be graphed to show peaks (highs) and troughs (lows) which correspond to the maximum and minimum values.

Since sinusoidal functions are periodic, they repeat their values in regular intervals. This repetition is a key feature that makes them valuable for modeling repetitive natural processes.

Amplitude and Midline

Amplitude and midline are two crucial aspects of sinusoidal functions that describe the extent and position of the wave's oscillations.

• Amplitude: In our equation, the amplitude is 15. This indicates how far the temperature moves above and below the average value (midline). It tells us the extreme points of temperature variation, spanning a total range of 30 degrees (from -15 to +15, considering both directions).
• Midline: The midline is the horizontal line around which the oscillations of the sinusoidal function occur. For this temperature function, it is set at \( y = 20 \). This means the average temperature around which the daily fluctuations occur is 20°F.

Understanding the amplitude and midline allows us to anticipate the maximum and minimum points of the graph, helping us interpret real-life implications, such as daily temperature shifts.

Frequency and Period

Frequency and period are important parameters that determine how quickly a sinusoidal function repeats its cycle.

• Frequency: In the equation \( y = 20 + 15 \sin \frac{\pi}{12} t \), the frequency is indicated by the term \( \frac{\pi}{12} \). This determines how fast the wave oscillates. A higher frequency would mean more cycles in a given time interval.
• Period: The period of a sinusoidal function is the time it takes to complete one full cycle. It is calculated as \( \frac{2\pi}{\text{frequency}} \). For this function, substituting \( \frac{\pi}{12} \) gives a period of 24 hours. This means the temperature cycle is completed once every 24-hour period, matching the natural daily cycle of temperatures.

The relationship between frequency and period illustrates the tempo of the oscillations. Grasping these helps understand how frequently the temperature will reach its highs and lows over a given time frame.