A certain flagpole that is 273 feet tall casts a shadow 130 feet long. Find the angle of elevation of the sun.
Understand the Problem
The problem asks to determine the angle of elevation of the sun given the height of a flagpole and the length of its shadow. This involves using trigonometric relationships, specifically the tangent function, to relate the angle to the ratio of the height and the shadow length. We will use the arctangent function (inverse tangent) to find the angle, and round the result to the nearest degree.
Answer
The angle of elevation of the sun is $65$ degrees.
Answer for screen readers
The sun's angle of elevation is $65$ degrees.
Steps to Solve
- Identify the trigonometric relationship
The angle of elevation, the height of the flagpole, and the length of the shadow form a right triangle, where the height is the opposite side, and the shadow is the adjacent side to the angle of elevation. The tangent function relates the angle to the opposite and adjacent sides:
$tan(\theta) = \frac{opposite}{adjacent}$
- Set up the equation
In this problem:
$opposite = 273$ feet (height of flagpole) $adjacent = 130$ feet (length of shadow)
So, $tan(\theta) = \frac{273}{130}$
- Solve for $\theta$
To find the angle of elevation $\theta$, take the arctangent (inverse tangent) of both sides of the equation:
$\theta = arctan(\frac{273}{130})$
- Calculate the arctangent
Using a calculator:
$\theta \approx 64.51$ degrees
- Round to the nearest degree
Rounding $64.51$ to the nearest degree gives $65$ degrees.
The sun's angle of elevation is $65$ degrees.
More Information
The angle of elevation is the angle between the horizontal plane and a line-of-sight, from the observer to some point of interest above the horizontal plane.
Tips
A common mistake is using the wrong trigonometric function (e.g., sine or cosine instead of tangent). Another is forgetting to use the arctangent function to find the angle. Finally, not rounding to the nearest degree as requested, or rounding incorrectly, can lead to an incorrect answer.
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