Question

USING TOOLS You use a video camera to pan up the Statue of Liberty. The height \(h\) (in feet) of the part of the Statue of Liberty that can be seen through your video camera after time \(t\) (in seconds) can be modeled by \(h=100 \tan \frac{\pi}{36} t\). Graph the function using a graphing calculator. What viewing window did you use? Explain.

Short answer

The function has been successfully graphed on the calculator. The viewing window that has been chosen is from 0 to 36 for t (time in seconds) and an appropriate range for h (height in feet) that allows for the visualization of the height changing over time.

Step-by-step solution

Understanding the function

The given function is \(h=100 \tan \frac{\pi}{36} t\). This function describes how the height of the part of the Statue of Liberty that can be seen through your video camera changes over time. The multiplier 100 adjusts the amplitude of the standard tangent function, and the fraction \(\frac{\pi}{36} \) changes the period of the function (i.e., how quickly it cycles).

Graphing the function

Use a graphing calculator to graph the function. Enter the function \(h=100 \tan \frac{\pi}{36} t\) into the graphing calculator. Make sure your calculator is set to radian mode since the argument of the tangent function is in radians.

Choosing the viewing window

The viewing window of a graph is the range of x and y values that are displayed. When choosing the viewing window, select a range for t (x-axis) that makes sense for the scenario and a range for h (y-axis) that allows you to see the behavior of the function. In this scenario, it makes sense to set a window for t from 0 to 36 seconds, which is a full period for this function. For h, you can start from 0 and select an upper range large enough to accommodate the peak value of the function.

Key concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is a fundamental trigonometric function that relates an angle in a right-angled triangle to the ratios of two of its sides. Specifically, it is the ratio of the length of the opposite side to the length of the adjacent side. In the context of graphing, the tangent function creates a curve that is periodic and has discontinuities – where the function goes to infinity, known as asymptotes.

When graphing the function, one key feature to observe is its periodicity, which is how often the pattern repeats. For a standard tangent function, this occurs every \( \pi \) radians. Changes to the function, like the multiplier and coefficient seen in our textbook example, will adjust the steepness (amplitude) and the rate at which the function repeats (period), respectively.

Understanding the Exercise

In the exercise, the height \( h \) is modeled by \( h=100 \tan \frac{\pi}{36} t \). With 100 scaling the standard tangent function, we can anticipate larger values for the output. The role of \( \frac{\pi}{36} \) is to extend the period of the tangent function, making the Statue of Liberty's height cycle through its values more slowly in relation to time. It's essential to recognize that as the angle approaches \( \frac{\pi}{2} \) and \( -\frac{\pi}{2} \), the function's value will approach infinity, resluting in those aforementioned vertical asymptotes.

Graphing Calculator Usage

A graphing calculator is an invaluable tool for visualizing trigonometric functions, like the tangent function in our exercise. To graph a trigonometric function, you usually need to input the function into the calculator in the correct syntax. In this case, you would enter \( h=100 \tan \frac{\pi}{36} t \) as given in the exercise.

Using graphing calculators simplifies the process of understanding complex functions, allowing for the quick visualization of graphs, manipulation of viewing windows, and even tracing points to better understand the function's behavior over a given domain.

Exercise Specifics

For the case at hand where the height of the Statue of Liberty is being modeled, the window setting is crucial. The selected window should cover one full period of the function and a sufficient range to visualize the peaks and troughs clearly. The optimal window for time (t) taken into account in our exercise is 0 to 36 seconds, making sure the entire behavior of function within one period is captured on the x-axis.

Radian Mode

Radian mode is a crucial setting when working with trigonometric functions on a graphing calculator. This mode is used because trigonometric functions are inherently linked to the concept of a circle, with radians being a natural way to measure angles in relation to the radius of that circle.

In radian mode, the values along the trigonometric functions correspond to the radians, which is different from degrees. There are \( 2\pi \) radians in a full circle, as opposed to 360 degrees. This means that for any trigonometric calculations or graphing exercises involving radians, it's essential to ensure that the calculator is switched to radian mode to accurately reflect the function's behavior.

Application in the Exercise

The coefficient \( \frac{\pi}{36} \) in our textbook problem is given in radians, thus requiring the calculator to be set to radian mode for accurate plotting. If the calculator were mistakenly left in degree mode, the graph would not correctly represent the function's behavior over time, and the understanding of the relationship between time and the height of the Statue of Liberty would be skewed.