Question

An aircraft (at Z ) is spotted by two observers (at X and Y ) who are L=1350 feet apart. As the airplane passes over the line joining them each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. If A=50 ∘ , and B=30 ∘ , how high is the airplane? The elevation of the plane is approximately feet. (Do not round until the final answer. Then round to two decimal places as needed.)

Answer 1

To find the height of the airplane, we use trigonometric functions to set up a system of equations based on the angles of elevation and the distance between observers. Solving these equations yields the airplane's elevation.

Step-by-step explanation:

The question involves finding the height of an airplane based on the angles of elevation measured by two observers a known distance apart. We can use trigonometry to solve this problem, specifically the tangent function which relates the angle of elevation to the height of the object above the ground.

To find the height of the airplane, we will first draw two right-angled triangles sharing the height as their opposite side. Observer X will form one triangle with angle A (50°) and observer Y will form the second triangle with angle B (30°). The distance L (1350 feet) between the observers will act as the adjacent side for the respective triangles.

L can be split into two parts, LX and LY, where LX is the distance from observer X to the point right below the plane, and LY is from observer Y to the same point. Using the tangent of the angles:

• Tan A = Height / LX
• Tan B = Height / LY

Since LX + LY = L, we have a system of two equations:

• Tan(50°) = Height / LX
• Tan(30°) = Height / (1350 - LX)

By solving these equations for LX and the Height, we determine the elevation of the plane.

Assuming no errors made in calculations, the height will be approximately:

Height = (Tan(30°) * Tan(50°) * L) / (Tan(50°) - Tan(30°))