To solve the problem, we can use trigonometry, specifically the tangent function, which relates the angle of elevation, the height of an object, and the distance from the object.
Here's how to solve it step by step:
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Drawing the Diagram:
- Imagine a right triangle where:
- The vertical leg represents the height of the blimp, which is 400 feet.
- The horizontal leg represents the distance from the fan to the stadium, which we will call \( d \).
- The angle of elevation from the fan's position to the blimp is 7 degrees.
- Imagine a right triangle where:
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Identifying the Trigonometric Ratio:
- We can use the tangent function, which is defined as: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
- In this case, the opposite side is the height of the blimp (400 feet), and the adjacent side is the distance \( d \) from the fan to the stadium.
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Setting Up the Equation:
- Replacing the terms in the tangent ratio: \[ \tan(7^\circ) = \frac{400}{d} \]
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Solving for \( d \):
- Rearranging the equation to solve for \( d \): \[ d = \frac{400}{\tan(7^\circ)} \]
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Calculating the Value:
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We will calculate \( \tan(7^\circ) \) using a scientific calculator or a trigonometric table: \[ \tan(7^\circ) \approx 0.1228 \]
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Now substitute \( \tan(7^\circ) \) into the equation: \[ d = \frac{400}{0.1228} \approx 3251.63 \text{ feet} \]
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Rounding to the Nearest Foot:
- Rounding 3251.63 to the nearest foot, we get: \[ d \approx 3252 \text{ feet} \]
Final Answer: The fan is approximately 3252 feet away from the stadium.
Summary of Work:
- Draw a right triangle with height (400 ft) and distance (d).
- Use the tangent function, set up the equation: \( \tan(7^\circ) = \frac{400}{d} \).
- Solve for \( d \) giving \( d \approx 3252 \) feet.
- Rounding to the nearest foot gives 3252 feet.