91Ó°ÊÓ

A right triangle is a type of triangle that has one angle equal to 90 degrees. The right angle creates two sides known as the base and the perpendicular, and the stretched side across from the right angle is the hypotenuse. This unique property makes right triangles critical in many mathematical calculations, like those involving the Pythagorean theorem.
In the context of our airplane problem, the plane flying at an altitude of 1 mile forms a right triangle with its flight path and the radar station on the ground. Here, the altitude of 1 mile is one side of the triangle, and the horizontal distance flown by the plane is the other. These two form the legs of the triangle, while the straight-line distance between the plane and the radar station represents the hypotenuse. This geometric relationship becomes essential for calculating distances accurately.

The distance function often involves the Pythagorean theorem when dealing with right triangles. According to this theorem:

• If you know the lengths of both legs of a right triangle, you can easily calculate the hypotenuse.
• For example, the formula used is: \( s^2 = d^2 + h^2 \).
• Here, "s" represents the hypotenuse (distance from radar to plane), "d" represents the horizontal distance the plane travels, and "h" is the altitude of the plane.

This theorem allows us to express the distance \(s\) as a function of \(d\) in our problem. Specifically, since the altitude is constant at 1 mile, we have \( s = \sqrt{d^2 + 1} \). It shows how the direct distance increases as the plane flies further horizontally, adhering to the right triangle's properties.

Horizontal distance is crucial when analyzing the motion of an object flying parallel to the ground. In practical terms, it's how we measure how far the plane has flown from its starting point directly above the radar station. This term is vital because it forms one of the sides of our right triangle.
In our scenario, at a constant speed of 350 miles per hour, the plane's horizontal distance \(d\) after \(t\) hours can be determined by a simple formula: \(d = 350t\). This relationship is straightforward due to the consistent speed, telling us that every hour the plane travels 350 miles horizontally. By knowing \(d\), we can use it as part of the input to determine the straight-line distance \(s\) with the help of the Pythagorean theorem.