Question

A skier is skiing through a series of moguls down a course that is \(200 \mathrm{m}\) in length at a constant speed of \(1 \mathrm{m} / \mathrm{s}\). The constant slope of the hill is -1. a) Write a function representing the skier's distance, \(d,\) from the base of the hill versus time, \(t,\) in seconds (neglecting the effects of the moguls). b) If the height, \(m,\) of the skier through the moguls, ignoring the slope of the hill, is \(m(t)=0.75 \sin 1.26 t,\) write a function that represents the skier's actual path of height versus time. c) Graph the function in part a) and the two functions in part b) on the same set of axes.

Short answer

a) \(d(t) = 200 - t\). b) \(H(t) = -t + 0.75 \sin(1.26t)\). c) Graph \(d(t)\), \(m(t)\), and \(H(t)\) from 0 to 200 seconds.

Step-by-step solution

Understanding the Problem

A skier is skiing down a hill with a slope of -1 at a constant speed of 1 m/s over a course of 200m. We need to find the skier's distance from the base of the hill as a function of time, the skier's height due to moguls, and graph these functions.

- Write function for distance versus time

Given the skier's speed is constant at 1 m/s, the distance function with respect to time can be written as a linear function: \[ d(t) = 200 - t \]. This represents the distance from the base of the hill where the skier starts from a distance of 200m and moves towards the base at a constant speed.

- Write the height function with moguls

The height function due to the moguls is given as: \[ m(t) = 0.75 \sin(1.26t) \]. To account for the overall path including the slope of the hill, the height function can be modified to include the slope factor. If the hill has a constant slope of -1, the height due to the slope will change linearly as: \[ h(t) = -t \]. Thus, the total height function will be: \[ H(t) = -t + 0.75 \sin(1.26t) \].

- Graph the Functions

To graph the functions: 1. Graph the distance function \[ d(t) = 200 - t \]. 2. Graph the height function due to the moguls \[ m(t) = 0.75 \sin(1.26t) \]. 3. Graph the total height function \[ H(t) = -t + 0.75 \sin(1.26t) \]. All functions should be plotted over the interval from \( t = 0 \) to \( t = 200 \) seconds.

Key concepts

Distance Function

A distance function describes how far an object is from a specific point over time. In our exercise, the distance function tells us how far the skier is from the base of the hill. Given that the skier is skiing down a hill with a constant speed of 1 meter per second, we can form the distance function.

Since the skier starts 200 meters away from the base and moves towards it at a speed of 1 meter per second, the distance function, denoted as d(t), where t is the time in seconds, becomes:

\( d(t) = 200 - t \)

This equation effectively means that for every second that passes, the distance to the base decreases by 1 meter. Hence, after 1 second, the skier is 199 meters away; after 2 seconds, the skier is 198 meters away, and so on. This is a straightforward linear function.

Height Function

A height function usually describes how an object's height changes over time. In this problem, the skier encounters moguls, which are bumps on the ski slope that affect their height. The height due to the moguls is given by a sinusoidal function:

\( m(t) = 0.75 \sin(1.26t) \)

This equation means that the height changes in a wave-like pattern with an amplitude of 0.75 meters and a frequency determined by 1.26 rad/s. The sine function is periodic and brings in the up-and-down nature of the moguls the skier experiences.

Sine Function

The sine function is an essential part of trigonometry. It varies periodically, between -1 and 1, creating a wave-like pattern. In the context of our skier, the function \( \sin(1.26t) \) influences the height of the skier to account for the moguls.

Here, 1.26 is the coefficient that affects the frequency of the sine wave. The '1.26t' means the wave completes its cycle more rapidly.
Understanding sine functions and their characteristics is crucial because they help model periodic phenomena in real life, like sound waves, light waves, and in this case, the bumpy ski path.

Graphing Functions

Graphing functions helps visualize how different variables in an equation interact over time. To fully understand our exercise, we graph three functions:

• The distance function \( d(t) = 200 - t \)
• The height function due to moguls \( m(t) = 0.75 \sin(1.26t) \)
• The total height function, combining slope and moguls: \( H(t) = -t + 0.75 \sin(1.26t) \)

When graphing these:

• For \( d(t) = 200 - t \), plot points from t=0 to t=200 to show a straight line decreasing evenly.
• For \( m(t) = 0.75 \sin(1.26t) \), note the wave pattern that oscillates between -0.75 and 0.75 meters.
• For \( H(t) = -t + 0.75 \sin(1.26t) \), plot the combined effects. You will observe a downward slope superimposed with the sine wave, showing the skier's actual path down the hill over time.

Visualization clarifies how these functions interact, making it easier to understand the skier's movement.