Question

Golden, British Columbia, is one of the many locations for heliskiing in Western Canada. When skiing the open powder, the skier leaves behind a trail, with two turns creating one cycle of the sinusoidal curve. On one section of the slope, a skier makes a total of 10 turns over a \(20-s\) interval. a) If the distance for a turn, to the left or to the right, from the midline is \(1.2 \mathrm{m}\) determine the function that models the path of the skier. b) How would the function change if the skier made only eight turns in the same 20-s interval?

Short answer

Function for Part a: \(y(t) = 1.2 \times \text{sin}(0.5πt) \). Function for Part b: \(y(t) = 1.2 \times \text{sin}(0.4πt) \).

Step-by-step solution

Determine the frequency of the skier's path (Part a)

The skier makes 10 turns over a 20-second interval. Each turn represents half a cycle, so for 10 turns, there are 5 complete cycles. The frequency is given by \(\frac{\text{number of cycles}}{\text{total time}} \). Therefore, the frequency is \(\frac{5 \text{ cycles}}{20 \text{ s}} = 0.25 \text{ cycles/s} \).

Determine the angular frequency

Angular frequency (ω) is related to frequency (f) by the formula \(\text{ω} = 2πf \). Thus, \(\text{ω} = 2π \times 0.25 = 0.5π \text{ rad/s} \).

Determine the amplitude

The distance for a turn, to the left or to the right, from the midline is given as 1.2 meters. Therefore, the amplitude (A) of the sine function modeling the path is \(\text{A} = 1.2 \text{ m} \).

Form the sinusoidal function (Part a)

The general form of a sinusoidal function is \(y(t) = A \times \text{sin}(ωt) \). Substituting the values for amplitude and angular frequency, the function becomes: \(y(t) = 1.2 \times \text{sin}(0.5πt) \).

Determine the frequency of the skier's path (Part b)

If the skier makes 8 turns over a 20-second interval, there are \(\frac{8}{2} = 4 \) complete cycles. The frequency in this case is \(\frac{4 \text{ cycles}}{20 \text{ s}} = 0.2 \text{ cycles/s} \).

Determine the angular frequency for Part b

For the new frequency, the angular frequency (ω) is \(\text{ω} = 2π \times 0.2 = 0.4π \text{ rad/s} \).

Form the sinusoidal function for Part b

Using the same amplitude (A = 1.2 m) and the new angular frequency, the function becomes: \(y(t) = 1.2 \times \text{sin}(0.4πt) \).

Key concepts

Frequency

When we talk about frequency in sinusoidal functions, we refer to the number of complete cycles or oscillations per unit time. In the context of our skiing example, frequency tells us how often the skier completes turns in a given time frame. If the skier makes 10 turns over 20 seconds, we calculate the number of complete cycles by noting that each turn represents half a cycle. Therefore, for 10 turns, there are 5 complete cycles. To find the frequency, we use the formula: \[\text{Frequency (f)} = \frac{\text{Number of Cycles}}{\text{Total Time}} = \frac{5}{20} = 0.25 \text{ cycles/s} \]This means the skier completes a full cycle 0.25 times every second.

Amplitude

Amplitude is a measure of how far the graph of a function deviates from its central axis. In our skiing scenario, amplitude reflects the maximum distance the skier veers to either the left or right from the midline. This distance helps define the intensity or maximum extent of the skier's movement. Given the skier turns 1.2 meters to the left or right, the amplitude (A) is 1.2 meters. So, the mathematical amplitude of our sinusoidal model is: A = 1.2 m.

Angular Frequency

Angular frequency, denoted by ω, links the linear frequency to the concept of angles and radians in trigonometric functions. It helps us understand how fast an oscillation occurs in terms of radians per second. The formula to convert frequency (f) to angular frequency (ω) is: \[ \text{ω} = 2πf \]For our skier, where the frequency is 0.25 cycles per second, the angular frequency becomes: \[ \text{ω} = 2π \times 0.25 = 0.5π \text{ rad/s} \]This value means the skier's path follows a cycle rate of 0.5π radians every second. If the frequency changes, such as in Part b of the problem (8 turns over 20 seconds resulting in 0.2 cycles/s), the angular frequency becomes: \[ \text{ω} = 2π \times 0.2 = 0.4π \text{ rad/s} \]

Sinusoidal Modeling

Sinusoidal modeling uses sine or cosine functions to represent periodic phenomena. These functions are valuable for describing repetitive patterns like the skier’s trail. The general form of a sinusoidal function is: \[ y(t) = A \times \text{sin}(ωt) \]Here, A is the amplitude and ω the angular frequency. Using the values we have determined: For Part a (10 turns in 20 seconds), the sinusoidal function is: \[ y(t) = 1.2 \times \text{sin}(0.5πt) \]For Part b (8 turns in 20 seconds), it is: \[ y(t) = 1.2 \times \text{sin}(0.4πt) \]This modeling provides a mathematical representation of the skier’s path, letting us predict their trajectory at any given moment. Understanding sinusoidal modeling is key in fields like engineering, physics, and even everyday contexts where cyclic behavior occurs.