How to Calculate the Height of a Plane using Trigonometry

What is the method to find the height of a plane using trigonometry based on the given scenario? The height of the plane can be determined by utilizing trigonometry principles, specifically the tangent function, in a scenario where observers on the ground at two different points spot the plane flying on a straight course. The angle of elevation from each observer's point is provided, along with the distance between the observers. By setting up two right triangles and using the tangent function for each angle of elevation, a system of equations can be formed to solve for the height of the plane.

Trigonometry and Height Calculation

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In this scenario, trigonometry is employed to find the height of the plane based on the angles of elevation observed by two observers on the ground.

Utilizing the Tangent Function

The tangent function is defined as the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle in a right triangle. By setting up two separate right triangles for each observer's point of view and applying the tangent function to the known angles of elevation, we can create equations to solve for the height of the plane.

Solving for the Height of the Plane

To find the height of the plane, we can set up the following equations based on the given angles of elevation:

Equation 1: tan(46°) = h / 2 (First observation point)

Equation 2: tan(71°) = h / 2 (Second observation point)

By solving this system of equations simultaneously, we can determine the height of the plane by averaging the calculated values obtained from both observation points. The final answer is approximately 3.20 miles, indicating the height of the plane in this scenario.

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