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Magazine Chapter 7: Problem 31
After a wind storm, a farmer notices that his 32 -foot windmill may be leaning, but he is not sure. From a point on the ground 31 feet from the base of the windmill, he finds that the angle of elevation to the of the windmill is \(48^{\circ}\). Is the windmill leaning? If so, what is the acute angle the windmill makes with the ground?
Short Answer
Expert verified
Yes, the windmill is leaning. The acute angle it makes with the ground is approximately \(65.1^{\circ}\).
Step by step solution
01
Understand the Problem
We have a windmill that is supposedly leaning. It originally has a height of 32 feet. From a point on the ground, 31 feet away from the base, the angle of elevation to the top of the windmill is measured to be \(48^{\circ}\). We need to determine if the windmill is actually leaning by comparing the expected vertical height with the given vertical height from the angle of elevation.
02
Set up a Right Triangle
Imagine a right triangle where the base is the distance from the point on the ground to the base of the windmill (31 feet), the height is the vertical height of the windmill (let's denote this as \(h\)), and the hypotenuse can be calculated using trigonometric ratios. We are told the angle of elevation is \(48^{\circ}\).
03
Use the Tangent Function
The tangent function relates the opposite side (the height, \(h\)) to the adjacent side (the distance from the point to the base of the windmill). Therefore, the tangent of the \(48^{\circ}\) is given by \(\tan(48^{\circ}) = \frac{h}{31}\).
04
Calculate the Height from Trigonometry
Using the tangent function, solve for \(h\):\[ h = 31 \times \tan(48^{\circ}) \]Calculate \(\tan(48^{\circ})\) using a calculator, which approximately equals 1.1106, then multiply:\[ h = 31 \times 1.1106 \approx 34.429 \text{ feet} \]
05
Compare to the Original Height
The calculated height \(34.429\) feet is different from the actual height \(32\) feet of the windmill, confirming that the windmill is leaning. The windmill should be 32 feet if perfectly vertical, but the angle of elevation suggests otherwise.
06
Determine the Leaning Angle
To find the angle the windmill makes with the ground, we use the cosine rule for the angle \(\theta\) of the leaned windmill. This angle can be found using cosine as:\[ \cos(\theta) = \frac{31}{34.429} \]Calculate \(\theta\):\[ \theta \approx \cos^{-1}\left(\frac{31}{34.429}\right) \approx 24.9^{\circ} \]
07
Calculate the Complementary Acute Angle
The acute angle that the windmill makes with the ground is \(90^{\circ} - \theta\):\[ 90^{\circ} - 24.9^{\circ} = 65.1^{\circ} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angle of Elevation
The angle of elevation is a key concept in trigonometry. It refers to the angle between the horizontal line and the line of sight looking upward to an object. Imagine you're standing on the ground looking up at the top of a windmill; this angle of elevation is what you would measure above the horizontal line. In practical terms, when measuring this angle, ensure your measuring device (like a protractor) is level with the horizon to get an accurate reading.
- An angle of elevation helps in determining the height of an object when you know the distance from the object.
- It is commonly used in problems involving heights and distances, such as measuring the height of a tree, a building, or in this case, a windmill.
Tangent Function
The tangent function is one of the fundamental trigonometric functions, often abbreviated as 'tan'. It relates to the angles and sides of right triangles, specifically:
- In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the adjacent side.
- Mathematically, for an angle \( \theta \): \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
Right Triangle
A right triangle is a triangle in which one of the angles is exactly 90 degrees. This triangle type is rich in geometric properties, making it highly valuable in trigonometry and practical applications alike.
- The two sides forming the right angle are called the 'legs', and the side opposite the right angle is the 'hypotenuse'.
- Right triangles are often the basis for defining the trigonometric functions: sine, cosine, and tangent.
- The base, known here as the distance from the measurement point to the windmill base (31 feet).
- The height, or opposite side, which is the effective vertical height of the windmill in our problem.
- The hypotenuse, which can be determined once the effective height is calculated to assess the windmill's condition.
Cosine Rule
The cosine rule is an extension of the basic trigonometric principles to non-right triangles. However, it is also useful in right triangles for solving angles or sides when the necessary information isn't directly aligned with the primary trigonometric functions.
- In a general triangle, the cosine rule states:\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \], where \( a \), \( b \), and \( c \) are the sides of the triangle, and \( C \) is the angle opposite side \( c \).
- In the windmill problem, we use part of this rule to find the angle related to the hypotenuse when we know it is not initially a 90 degree angle due to the lean.
