Problem 56 - Algebra and Trigonometry

Question

Length of a Shadow On a day when the sun passes directly overhead at noon, a six-foot-tall man casts a shadow of length $$S(t)=6\left|\cot \frac{\pi}{12} t\right|$$ where \(S\) is measured in feet and \(t\) is the number of hours since 6 A.M. (a) Find the length of the shadow at \(8 : 00\) A.M., noon, \(2 : 00 \mathrm{P}, \mathrm{M},\) and \(5 : 45 \mathrm{P.M}\) (b) Sketch a graph of the function \(S\) for \(0 < t < 12\). (c) From the graph determine the values of \(t\) at which the length of the shadow equals the man's height. To what time of day does each of these values correspond? (d) Explain what happens to the shadow as the time approaches 6 \(\mathrm{P} . \mathrm{M}\) . (that is, as \(t \rightarrow 12^{-} )\) .

Step-by-Step Solution

Answer

(a) 2√3 feet at 8 A.M., 0 feet at noon, 2√3 feet at 2 P.M., 18.46 feet at 5:45 P.M.; (c) Equal length at 9 A.M. and 3 P.M.; (d) Shadow approaches infinity at 6 P.M..

Step 1: Evaluate at 8:00 A.M.

Convert 8:00 A.M. to hours since 6 A.M.: \(t = 2\). Substitute into the formula: \[S(2) = 6 \left|\cot \frac{\pi}{12} \times 2\right| = 6 \left|\cot \frac{\pi}{6}\right| = 6 \times \frac{\sqrt{3}}{3} = 2\sqrt{3} \text{ feet} \]

Step 2: Evaluate at noon

Convert noon to hours since 6 A.M.: \(t = 6\). Substitute into the formula:\[S(6) = 6 \left|\cot \frac{\pi}{12} \times 6\right| = 6 \left|\cot \frac{\pi}{2}\right| = 6 \times 0 = 0 \text{ feet} \]

Step 3: Evaluate at 2:00 P.M.

Convert 2:00 P.M. to hours since 6 A.M.: \(t = 8\). Substitute into the formula:\[S(8) = 6 \left|\cot \frac{\pi}{12} \times 8\right| = 6 \left|\cot \frac{2\pi}{3}\right| = 6 \times \frac{1}{\sqrt{3}} = 2\sqrt{3} \text{ feet} \]

Step 4: Evaluate at 5:45 P.M.

Convert 5:45 P.M. to hours since 6 A.M.: \(t = 11.75\). Substitute into the formula:\[S(11.75) = 6 \left|\cot \left(\frac{\pi}{12} \times 11.75\right)\right| \]Calculate \(\cot\left(\frac{11.75\pi}{12}\right) \approx \cot\left(\frac{11\pi}{12} - \frac{3\pi}{48}\right)\), and solve for:\[S(11.75) = 6 \times 3.077 = 18.46 \text{ feet (approximately)}\]

Step 5: Sketching the Function Graph

Evaluate how \(S(t)\) behaves from \(t = 0\) to \(t = 12\). Note that- At \(t = 6\), the shadow is zero.- Symmetrical behavior around \(t = 6\). - Increases as it approaches \(t = 0\) and \(t = 12\) beyond \(6\) feet. Graph would be into infinity as \(t\) approaches \(t = 0^+\) and \(t = 12^-\).

Step 6: Find when Shadow Equals Height

Set \(S(t) = 6\) and solve:\[6 \left|\cot \frac{\pi}{12} t\right| = 6 \Rightarrow \left|\cot \frac{\pi}{12} t\right| = 1 \Rightarrow \frac{\pi}{12} t = \frac{\pi}{4}, \frac{3\pi}{4}; \text{ etc, }\]Solution for \(t\) results in \(t = 3, 9\). The shadow equals 6 feet at 9 A.M. and 3 P.M.

Step 7: Evaluate as Time Approaches 6 P.M.

As \(t \to 12^-\), \(6 \left|\cot \frac{\pi}{12} t\right|\) tends to a very large value because \(\cot\left(\frac{\pi}{12} \times 12\right)\) approaches infinity, meaning the shadow would be infinitely long.

Key Concepts

Cotangent Function

The cotangent function, often denoted as \( \cot(x) \), is one of the fundamental trigonometric functions. It is the reciprocal of the tangent function, meaning it can be expressed as \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \). This function is periodic, with a period of \( \pi \), which means it repeats its values over intervals of \( \pi \) radians. In simple terms, this means the cotangent function displays a repeating pattern every \(180^\circ\).
This function has some essential characteristics:

• It is undefined where the sine function is zero, such as at multiples of \( \pi \), since you cannot divide by zero.
• It decreases from positive to negative between its undefined points.
• The cotangent has vertical asymptotes at these undefined points, where the value of the function approaches infinity.

This behavior is important for understanding real-life applications, such as calculating the length of shadows cast by a vertical object depending on the time of day.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cotangent provides a visual representation of their behavior over time or angles. When graphing \( \cot(x) \), pay attention to the repeating pattern and undefined points, also called vertical asymptotes. These vertical asymptotes occur at the multiples of \( \pi \), which signifies breaks in the graph.
To graph trigonometric functions effectively, follow these steps:

• Identify the period, which for \( \cot(x) \) is \( \pi \).
• Mark the points where the function is undefined.
• Plot points in between these undefined points using known values, such as where \( \cot(\frac{\pi}{4}) = 1 \).
• Draw the curve that passes through these points, indicating how the function approaches and recedes from its asymptotes.

Understanding these graphs is helpful in real-life scenarios, such as measuring periodic phenomena like tidal waves or daylight hours.

Application of Trigonometric Functions in Real Life

Trigonometric functions have many real-world applications, enhancing various fields like architecture, navigation, and even in day-to-day problems. One such application is calculating the length of shadows based on the angle of the sun, which is critical in fields like surveying and solar panel installation.
Using trigonometric functions, we can:

• Determine the height of a distant object using only its shadow length and the angle of elevation, thanks to formulas involving the tangent or cotangent.
• Design and optimize solar panels for maximum efficiency by calculating the sun's path and the resulting shadow lengths throughout the day.
• Create architectural structures with strategic shadow designs for cooling purposes, as seen in sustainable building designs.

These applications demonstrate the practical use of trigonometry in predicting and optimizing real-world outcomes.

Cotangent Behavior

Understanding the behavior of the cotangent function is crucial for interpreting its impact on real-life applications. The cotangent is notable for its decreasing nature between its asymptotes, starting from a positive value and becoming negative as it crosses zero. It is essential to recognize that:

• As the angle approaches \( \pi \), \( \cot(x) \) approaches zero and becomes negative beyond this point.
• As the angle approaches a multiple of \( \pi \) from the left, \( \cot(x) \) climbs towards infinity, and from the right, it drops from negative infinity.
• These steep changes can represent phenomena where rapid transitions occur, such as the shifting length of a shadow as the sun moves across the sky.

By understanding how the cotangent function behaves, we can better interpret situations where rapid angular changes influence critical measures.

Trigonometric Equations

Solving trigonometric equations involves finding the values of angles that satisfy equations involving trigonometric functions. These equations are essential in scenarios where you need to understand periodic or changing phenomena. Consider a situation where you are asked to find when a shadow is equal to the height of the object casting it.
To solve an equation like \( 6 \left| \cot \left( \frac{\pi}{12} t \right) \right| = 6 \), first simplify it to \( \left| \cot \left( \frac{\pi}{12} t \right) \right| = 1 \).
The next steps involve:

• Identifying standard angles where \( \cot(\theta) = 1 \) or \( -1 \) (e.g., \( \frac{\pi}{4} \) or \( \frac{3\pi}{4} \)).
• Solving for \( t \) from these angles within the given parameters (like time constraints).
• Understanding that these solutions recur due to the periodic nature of trigonometric functions.

Tackling these problems gives insight into cyclical patterns and equips one with the skills to predict outcomes based on given circumstances.