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Part 1
A plane rises from​ take-off and flies at an angle of 14 degrees with the horizontal runway. When it has gained 500 ​feet, find the​ distance, to the nearest​ foot, the plane has flown.
14 degrees
c equals question mark


We have a trigonometric problem where a plane takes off and flies at an angle of 14 degrees to the horizontal. The problem asks us to find the distance the plane has flown when it has gained 500 feet of altitude.

This scenario forms a right triangle:
- The vertical leg is the altitude gained: \(500\) feet.
- The angle of ascent is \(14^\circ\).
- We are looking for the hypotenuse of this right triangle, which represents the distance flown.

Using trigonometry, we can apply the sine function because:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Here, the opposite side is the altitude gained (500 feet), and the hypotenuse is the distance flown, which we want to find.

Thus, the equation becomes:
\[
\sin(14^\circ) = \frac{500}{\text{distance}}
\]
Rearranging the equation to solve for the distance:
\[
\text{distance} = \frac{500}{\sin(14^\circ)}
\]

Let me calculate that for you.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can easily solve this using a calculator:
1. Find the sine of 14 degrees.
2. Divide 500 by the sine of 14 degrees to get the distance the plane has flown.

If you need further assistance, feel free to ask!

Would you like any additional explanations? Here are some related questions to explore:

1. How does the angle of ascent affect the distance traveled?
2. What is the formula for calculating horizontal distance in this scenario?
3. Can we use a different trigonometric function to solve this problem?
4. What is the importance of understanding right-angle triangles in flight dynamics?
5. How would the calculation change if the angle were 30 degrees?

**Tip:** Always ensure that angles are in the correct unit (degrees or radians) when using trigonometric functions in calculations.

A plane rises from take-off and flies at an angle of 14 degrees with the horizontal runway. When it has gained 500 feet, find the distance, to the nearest foot, the plane has flown.

Learn how to calculate the distance flown by a plane that ascends at an angle of 14 degrees after gaining 500 feet of altitude. This trigonometry problem is ideal for students in Grades 9-12 and involves using the sine function to find the hypotenuse of a right triangle.

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