Problem 179 - Calculus Volume 1

Question

An alternating current for outlets in a home has voltage given by the function \(V(t)=150 \cos 368 t,\) where \(V\) is the voltage in volts at time \(t\) in seconds. a. Find the period of the function and interpret its meaning. b. Determine the number of periods that occur when 1 sec has passed.

Step-by-Step Solution

Answer

1. Period is 0.0171 sec; it cycles in 0.0171 sec. 2. About 58 periods occur in 1 sec.

Step 1: Identify the Standard Form

The standard form of a cosine function is \( V(t) = A \cos(Bt) \), where \( A \) is the amplitude, \( B \) affects the period, and \( t \) is time.

Step 2: Determine the Period Formula

For the cosine function \( V(t) = A \cos(Bt) \), the period \( T \) is given by \( T = \frac{2\pi}{|B|} \).

Step 3: Substitute and Solve for the Period

Using the function \( V(t) = 150 \cos(368t) \), we identify \( B = 368 \). Substituting into the period formula, we get: \[ T = \frac{2\pi}{368}. \] Calculate \( T \) to find the period of the function.

Step 4: Calculate the Period

Perform the calculation: \[ T = \frac{2\pi}{368} \approx 0.0171 \text{ seconds} \]. Thus, the period of the voltage function is approximately 0.0171 seconds.

Step 5: Interpret the Period

The period of 0.0171 seconds means the voltage function completes one full cycle in approximately 0.0171 seconds.

Step 6: Calculate Number of Periods in One Second

To find how many periods fit into one second, divide 1 second by the period found: \[ \text{Number of Periods} = \frac{1}{0.0171} \approx 58.48. \]

Step 7: Interpret the Number of Periods

Approximately 58.48 periods occur within one second, meaning the voltage function cycles about 58 times each second.

Key Concepts

Understanding Alternating Current

Alternating current (AC) is the type of electrical current typically used in homes and businesses for outlets. It differs from direct current (DC), where the flow of electric charge is unidirectional. AC repeatedly changes direction, represented graphically as a sine wave.

This oscillating nature of AC can be expressed using trigonometric functions such as the cosine function. By using these functions, the voltage at any given time can be accurately described, making it easier to predict and manage electrical systems.

Benefits of Alternating Current

• Efficient transmission over long distances: AC can be easily transformed to higher or lower voltages, which minimizes power loss over long distances.
• Compatibility with transformers: AC can be readily changed to different voltages using transformers, which makes it versatile for various applications.
• Widely used in residential and commercial systems: Almost all household and building electrical systems are based on AC.

Exploring the Cosine Function

The cosine function is a key component in trigonometry with applications far beyond geometry.In physics and engineering, it’s often used to model periodic phenomena like sound waves, light waves, and electrical currents such as alternating current.

Mathematically, the cosine function is defined for an angle, but in the context of AC circuits, we use it to relate time to voltage or current. The general form of a cosine function is given by:

\( V(t) = A \cos(Bt) \)

where

• \( A \) represents the amplitude, or the peak value of the wave
• \( B \) relates to the period of the wave
• \( t \) is the time variable

Characteristics of the Cosine Function

• Periodicity: The function repeats its values in regular intervals determined by the period.
• Symmetry: Cosine is an even function, meaning \( \cos(-t) = \cos(t) \).
• Amplitude: The height from the centerline to the peak of the wave.

Understanding the Period of a Function

The period of a function, especially trigonometric ones like the cosine function, is the length of time it takes for the function to complete one full cycle.For AC voltages defined by a cosine function, the period indicates how quickly the voltage oscillates.

To find the period \( T \) of the function \( V(t) = A \cos(Bt) \), we use the formula:\[ T = \frac{2\pi}{|B|} \] The smaller the period, the more cycles occur in a given time frame, like a second.

Interpreting the Period

• The period is vital for understanding the frequency of an AC system.
• A short period means higher frequency and more oscillations per second.
• In our example, the period of approximately 0.0171 seconds indicates a high frequency, which is typical for standard electrical systems operating at 60 Hz.