Problem 36 - Algebra 2

Question

The period of a periodic function is 8 s. How many cycles does it go through in 30 \(\mathrm{s?}\) F. \(\frac{4}{15}\) cycle G. 3.75 cycles H. 22 cycles J. 240 cycles

Step-by-Step Solution

Answer

G. 3.75 cycles

Step 1: Calculate Frequency

First, calculate the frequency of the periodic function by taking the reciprocal of its period. Given that the period of the function is 8 seconds, its frequency is \(1/8 = 0.125\) cycles per second.

Step 2: Calculate Number of Cycles in 30 seconds

Next, to find how many cycles the function goes through in 30 seconds, multiply the frequency of the function (0.125 cycles per second) by the total time (30 seconds). The number of cycles is therefore \(0.125 \times 30 = 3.75\) cycles.

Key Concepts

Frequency

Frequency tells us how often something repeats over a specific time period. It is an essential concept when studying periodic functions, like oscillations or waves. For example, a pendulum swings back and forth with a particular frequency. Frequency is defined as the number of cycles that occur in one second.

• It is usually measured in hertz (Hz), where 1 Hz equals 1 cycle per second.
• In this exercise, the frequency was found by taking the reciprocal of the period.

So, if a function repeats every 8 seconds, its frequency is the reciprocal, which is 0.125 Hz or 0.125 cycles per second. Understanding frequency helps us determine how quickly or slowly a function completes its cycles over time.

Cycle Calculation

Calculating the number of cycles in a given time frame is important when analyzing periodic functions. This involves two steps: finding the frequency and then determining how many cycles occur over a specific duration.

• First, determine the frequency of the function, which is how many cycles occur per second.
• Next, multiply this frequency by the total time of interest to find the number of complete cycles.

For instance, in our exercise, we have a frequency of 0.125 cycles per second. To find out how many cycles occur in 30 seconds, simply multiply: 0.125 cycles/second × 30 seconds = 3.75 cycles. This calculation shows that the function completes 3.75 cycles in 30 seconds.

Period

The period of a periodic function is the duration of time it takes to complete one full cycle. It tells us how long a repeating activity lasts before it starts over again.

• The period is the inverse of the frequency: if the period is known, the frequency can be calculated as the reciprocal.
• A shorter period means the function cycles more frequently, whereas a longer period indicates less frequent cycles.

Given a period of 8 seconds in the exercise, it means each cycle of the function takes 8 seconds to complete. This information is crucial for calculation and for understanding the timing of periodic events.