To find the horizontal distance \( x \) that the plane has flown when it is at an altitude of 250 feet and taking off at a 15° angle, we can use the properties of right triangles.
We can use the sine function, which relates the angle of the triangle to the ratio of the opposite side (altitude) and the hypotenuse (the distance the plane has flown).
The sine function is defined as: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this case:
- \(\theta = 15^\circ\)
- The opposite side (altitude) = 250 feet
Let \( d \) be the hypotenuse, or the distance that the plane has flown at that altitude. Using the sine function, we have: \[ \sin(15^\circ) = \frac{250}{d} \]
We can rearrange this to find \( d \): \[ d = \frac{250}{\sin(15^\circ)} \]
Calculating \(\sin(15^\circ)\): \[ \sin(15^\circ) \approx 0.2588 \]
Now substitute: \[ d = \frac{250}{0.2588} \approx 967.77 \text{ feet} \]
Now we can find \( x \), the horizontal distance traveled, using the cosine function: \[ \cos(15^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
The adjacent side is \( x \) (the horizontal distance): \[ \cos(15^\circ) = \frac{x}{d} \]
Now rearrange to solve for \( x \): \[ x = d \cdot \cos(15^\circ) \]
Calculating \(\cos(15^\circ)\): \[ \cos(15^\circ) \approx 0.9659 \]
Now substitute for \( d \): \[ x = 967.77 \cdot 0.9659 \approx 933.32 \text{ feet} \]
Finally, rounding to the nearest whole number, we find: \[ x \approx 933 \text{ feet} \]
Thus, the horizontal distance \( x \) that the plane has flown is approximately: \[ \boxed{933} \text{ feet} \]
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