Problem 57 - Algebra and Trigonometry

Question

Lighthouse The beam from a lighthouse completes one rotation every two minutes. At time \(t\) , the distance \(d\) shown in the figure on the next page is $$ d(t)=3 \tan \pi t $$ where \(t\) is measured in minutes and \(d\) in miles. (a) Find \(d(0.15), d(0.25),\) and \(d(0.45) .\) (b) Sketch a graph of the function \(d\) for \(0 \leq t<\frac{1}{2}\) (c) What happens to the distance \(d\) as \(t\) approaches \(\frac{1}{2} ?\)

Step-by-Step Solution

Answer

(a) \( d(0.15) \approx 1.39, d(0.25) = 3, d(0.45) \approx 30.61 \). (b) Sketch shows rapid increase towards infinity. (c) \( d(t) \to \infty \) as \( t \to \frac{1}{2} \).

Step 1: Evaluate Functional Value at Specific Points

To find the values of \( d(t) \) at specific points, simply substitute \( t = 0.15, 0.25, \) and \( 0.45 \) into the function \( d(t) = 3 \tan \pi t \). Calculate each value:\[\begin{align*}d(0.15) &= 3 \tan(\pi \times 0.15) = 3 \tan(0.15\pi),\d(0.25) &= 3 \tan(\pi \times 0.25) = 3 \tan(0.25\pi),\d(0.45) &= 3 \tan(\pi \times 0.45) = 3 \tan(0.45\pi).\end{align*}\]

Step 2: Substitute Values into the Tangent Function

Use a calculator to evaluate the tangent of the angles obtained in the previous step and multiply by 3:\[\begin{align*}d(0.15) &\approx 3 \times 0.46365 \approx 1.39095,\d(0.25) &\approx 3 \times 1 \approx 3,\d(0.45) &\approx 3 \times 10.20304 \approx 30.60912.\end{align*}\]

Step 3: Sketch the Graph of the Function

To sketch the graph of \( d(t) = 3 \tan \pi t \) from \( 0 \leq t < \frac{1}{2} \):- Note that the tangent function repeats every \( \pi \), thus experiencing asymptotes leading to vertical lines where \( t \to \frac{1}{2} \).- Plot the calculated points: \( (0.15, 1.39), (0.25, 3), \) and \( (0.45, 30.61) \).- Remember the behavior of \( \tan(\pi t) \): it drastically increases as \( t \to \frac{1}{2} \). Thus, \( d(t) \to \infty \) as \( t \to \frac{1}{2}^- \). Draw the graph accordingly, ensuring that it approaches infinity near \( t = 0.5 \).

Step 4: Analyze Asymptotic Behavior

As \( t \to \frac{1}{2} \), \( \pi t \to \frac{\pi}{2} \), thus: - \( \tan(\pi t) \to \tan\left(\frac{\pi}{2}\right) \to \infty \).- Therefore, \( d(t) = 3 \tan(\pi t) \to \infty \) as \( t \) approaches \( \frac{1}{2} \). This implies that the distance \( d \) increases without bound.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions and can be expressed as \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). It is a periodic function, meaning it repeats itself at regular intervals. Specifically, the tangent function has a period of \( \pi \), meaning it repeats every \( \pi \) units.

In our exercise, the distance \( d(t) \) is given by \( 3 \tan(\pi t) \). This means that for every complete revolution of the lighthouse beam (every \( 2 \) minutes), the tangent function resets its behavior. As you compute \( \tan(\pi t) \) at different values of \( t \), such as \( t = 0.15, 0.25, \text{ and } 0.45 \), you utilize the unique properties of the tangent function to understand the distance at those moments.

• The value of the function increases or decreases sharply around certain points due to its steep slope.
• The outputs can vary drastically, as seen in the extreme values calculated at different \( t \) values in our exercise.

Understanding how the tangent function operates is crucial for interpreting the results as \( t \) moves through its cycle.

Asymptotic Behavior

The asymptotic behavior of a function refers to the way the function behaves as it approaches a specific value where it tends to infinity or negative infinity. For the tangent function, this is a key characteristic. As \( t \) approaches certain values, the function \( \tan(\pi t) \) can become unbounded.

In this problem, as \( t \) approaches \( \frac{1}{2} \), the expression \( \pi t \) approaches \( \frac{\pi}{2} \). We know that \( \tan \left( \frac{\pi}{2} \right) \) becomes undefined or reaches infinity. Thus, \( \tan(\pi t) \) (and therefore \( d(t) = 3 \tan(\pi t) \)) tends to \( \infty \) as \( t \to \frac{1}{2} \).

• This is manifested on the graph as the function shooting upwards, representing an infinite distance.
• This behavior has practical significance, indicating that the lighthouse beam extends far away in one particular direction as time approaches the half-minute mark in this rotation context.

Understanding asymptotic behavior is vital for accurately sketching graphs and predicting the behavior of functions as they near critical points.

Function Evaluation

Function evaluation involves determining the value of a function at specific points. To evaluate \( d(t) = 3 \tan(\pi t) \) at specific values:

• Substitute \( t = 0.15, 0.25, \text{ and } 0.45 \) into the equation.
• Calculate \( \tan(\pi \times t) \) for each corresponding \( t \) value.

Using available tools or a calculator, extrapolate the specific values:

• For \( t = 0.15 \), \( d(0.15) \approx 1.39 \)
• For \( t = 0.25 \), \( d(0.25) = 3 \)
• For \( t = 0.45 \), \( d(0.45) \approx 30.61 \)

This exercise demonstrates the significant impact of input values on the resulting outputs. The tangent function’s sensitivity to changes emphasizes the need for precise calculations to capture the correct behavior at chosen time intervals.

Graph Sketching

Graph sketching is a powerful way to visualize how a function behaves over a specific interval.

To sketch the graph of \( d(t) = 3 \tan(\pi t) \) for \( 0 \le t < \frac{1}{2} \):

• First, consider plotting the points obtained from the function evaluation: \( (0.15, 1.39), (0.25, 3), \text{ and } (0.45, 30.61) \).
• Keep in mind the asymptotic behavior; the function approaches infinity as \( t \to \frac{1}{2} \).

Remember, for the tangent function, the graph will have a vertical asymptote at \( t = \frac{1}{2} \), depicted by a line the graph can approach but never touch or cross.

This will give you a curve that starts small, then rises steadily, shooting up steeply as it nears the asymptote. Visualizing through graph sketching helps in understanding the dynamics of the lighthouse beam’s distance as it varies over time. It brings to life the drastic changes that occur as you near critical points, such as \( t = \frac{1}{2} \), and the effects of trigonometric functions in real-world applications.