To determine the horizontal distance \( x \) the plane has flown when it reaches an altitude of 250 feet at an angle of 15°, we can use the sine function from trigonometry. The sine function relates the angle of a right triangle to the ratio of the opposite side (the altitude) to the hypotenuse (which is the actual distance the plane has traveled).
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We first have the following relationship: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{d} \] where \( h \) is the altitude (250 feet), \( d \) is the hypotenuse (the distance traveled by the plane), and \( \theta \) is the angle (15°).
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Since we want to find the horizontal distance \( x \), we can also relate \( x \) to the angle and the hypotenuse using the cosine function: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{d} \]
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First, we need to find the hypotenuse \( d \) using the sine function: \[ \sin(15°) = \frac{250}{d} \] Rearranging gives: \[ d = \frac{250}{\sin(15°)} \]
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Now we can calculate \( \sin(15°) \): \[ \sin(15°) \approx 0.2588 \quad (\text{value from calculator}) \]
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Now substitute \( \sin(15°) \) back into the equation for \( d \): \[ d = \frac{250}{0.2588} \approx 967.4 \text{ feet} \]
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Next, we can now find the horizontal distance \( x \) using the cosine function: \[ \cos(15°) = \frac{x}{d} \] Rearranging gives: \[ x = d \cdot \cos(15°) \]
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Now calculate \( \cos(15°) \): \[ \cos(15°) \approx 0.9659 \quad (\text{value from calculator}) \]
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Substitute \( d \) into the equation for \( x \): \[ x = 967.4 \cdot 0.9659 \approx 933.2 \text{ feet} \]
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Finally, rounding to the nearest whole number: \[ x \approx 933 \text{ feet} \]
Thus, the horizontal distance \( x \) the plane has flown is approximately \( 933 \) feet.
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