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Chapter 1: Problem 104

A pole of length \(L\) is carried horizontally around a corner where a 3 -ft- wide hallway meets a 4 -ft-wide hallway. For \(0<\theta<\pi / 2,\) find the relationship between \(L\) and \(\theta\) at the moment when the pole simultaneously touches both walls and the corner \(P .\) Estimate \(\theta\) when \(L=10 \mathrm{ft}\)

Short Answer

Expert verified
Answer: The relationship between the pole length L and the angle theta is given by the equation: \(L = 3 \sin \arctan(\frac{3}{4}) + 4 \sin (\pi / 2 - \theta)\). When L = 10 ft, the estimated value of theta is approximately 1.328 radians or 76.1 degrees.

Step by step solution

01

Draw a diagram of the situation

First, draw a diagram to better understand the problem. This includes the 3-ft and 4-ft hallways, the corner P, the angle \(\theta\), and the pole L touching both walls.
02

Create a right triangle in the corner

Create a right triangle in the corner with the 3-ft-wide hallway as one leg and the 4-ft-wide hallway as the other leg. Let A be the point where the pole touches the 3-ft hallway, B be the point where the pole touches the 4-ft hallway, and C be the intersection point of the hallways. Now, we have right triangle ABC with the right angle at C.
03

Use trigonometry to determine the angles of triangle ABC

We can use trigonometry to find the angles of triangle ABC. Since we are given the lengths of the legs, opposite to angles A and B, we can use tangent function. \(\tan A = \frac{3}{4}\) and \(\tan B = \frac{4}{3}\) To find the angles A and B, take the arctangent of the corresponding sides' ratios: \(A = \arctan(\frac{3}{4})\) and \(B = \arctan(\frac{4}{3})\)
04

Express angle \(\theta\) in terms of angles A and B

Since angle \(\theta\) is the angle between the pole and the vertical wall (4-ft hallway), we have: \(\theta = \pi / 2 - B\)
05

Find the length of the pole L

Now we can find the length of the pole by dividing its length into two parts: from corner P to A and from corner P to B. Since triangle ABC is a right triangle, we can use the sine function: \(L = PA + PB = 3 \sin A + 4 \sin B\)
06

Substitute the expressions for angles A, B, and \(\theta\) to find the relationship between \(L\) and \(\theta\)

Substitute the expressions for angles A and B from Step 3 and the expression for \(\theta\) from Step 4 to find the relationship between \(L\) and \(\theta\): \(L = 3 \sin \arctan(\frac{3}{4}) + 4 \sin (\pi / 2 - \theta)\)
07

Estimate the angle when \(L = 10 \mathrm{ft}\)

Plug in the given value of \(L = 10 \mathrm{ft}\) and solve for \(\theta\): \(10 = 3 \sin \arctan(\frac{3}{4}) + 4 \sin (\pi / 2 - \theta)\) We can solve for \(\theta\) numerically using a calculator or software: \(\theta \approx 1.328 \ \mathrm{radians}\) or \(\theta \approx 76.1^{\circ}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometry in Problem Solving
Geometry plays a huge role in solving complex problems like the exercise of carrying a pole around a corner in a hallway.
This discipline of mathematics helps us understand spatial relationships and shape properties, which are crucial when dealing with real-world scenarios.
In this specific problem, we aim to investigate how a physical object interacts with its environment using geometric principles.
Identifying key geometric elements, such as points, lines, and angles, can simplify problem-solving.
Creating a clear diagram as the problem solution suggests can visually represent abstract concepts like distances and angles in a tangible way.
By recognizing the right triangle forming where the hallways meet, we simplify many aspects of geometric analysis.
This helps in deducing relationships between different parts of the triangle and introduces trigonometric concepts to determine angles and distances accurately.
Right Triangle
A right triangle is at the heart of this problem's solution.
Right triangles have a special property: one of their angles is always 90 degrees.
In the context of the hallways problem, the corner formed by the two hallways is a right triangle.
In this right triangle, the two hallways represent the legs (3-ft and 4-ft).
The pole is a hypotenuse that interacts with these legs.
Knowing the lengths of two sides allows us to leverage trigonometry to find all the angles, specifically using
  1. The tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle.
With this, we could uncover angles A and B where the pole touches the respective hallways, crucial to further find the angle \(\theta\).
Angle Estimation
Estimating angles is integral in applications of trigonometry, like approximating \(\theta\) when the pole measures 10 feet in length. Initially, we figure out angles A and B using the arctangent function which translates the side ratio into angles.
The angle \(\theta\) is determined by knowing that it's complementary to angle B within the 90-degree half-plane formed by the vertical 4-ft wall.
\(\theta = \frac{\pi}{2} - B\), helps in setting a foundation for subsequent calculations.
To finalize the estimation process, note that solving for \(\theta\) involves substituting known values and solving numerically. This requires understanding small numerical approximations or using technology like calculators to get: \(\theta \approx 1.328\) radians or \(76.1^{\circ}\).

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