We are given that the searchlight is 6500 feet from a weather station, and we're asked to evaluate the height of the cloud ceiling if the angle of elevation to the spot of light on the clouds above the station is 45^(∘). Let's sketch a diagram describing this situation. We will call the height h.
As we can see, the spot of light, the horizontal distance, and the height form a 45^(∘)-45^(∘)-90^(∘) triangle. Let's recall that in this special right triangle the legs are congruent. Therefore, the height is also 6500 feet.
Alternative Solution
Solving using the tangent.
We can also find the height of the cloud ceiling without using the properties of 45^(∘)-45^(∘)-90^(∘) triangles. Let's recall the definition of the tangent. The tangent of∠Ais the ratio of the leg opposite∠Ato the leg adjacent∠A.
Using this definition, we can create an equation for tan 45^(∘).
tan 45^(∘)=h/6500
Let's solve the above equation.
