Lesson 3

You discussed angle of elevation with your partner in Share 1. Here is an example of solving a problem involving an angle of elevation.

 

Example

 

A hot-air balloon is 100 m higher than an observer on the ground. From the observer’s vantage point, the angle of elevation of the balloon is 37°. What is the straight-line distance from the observer to the balloon? Round your answer to the nearest metre.

 

Solution

 

Draw and label a diagram. Let x be the distance from the balloon to the observer.

 

 

From SOH-CAH-TOA, sine is the appropriate ratio.

 

 

The balloon is approximately 166 m from the observer.

 

Example

 

When Canada’s first transcontinental railroad was completed in 1885, the highest point on the line was at Kicking Horse Pass near Field, British Columbia. For trains headed west through the pass, there was a dangerous descent to the foot of Mt. Stephen. There the grade was 4.5%, which was more than four times the usual grade. The steep grade on Mt. Stephen, referred to as the Big Hill, led to numerous accidents.

 

A grade (slope) of 4.5% means the rail line rises 4.5 m for every 100 m it travels horizontally. With this grade, what is the angle that the rail line is inclined to the horizontal? Round your answer to the nearest tenth of a degree.

 

Solution

 

Sketch and label a diagram. Let θ be the angle of elevation.

 

 

From SOH-CAH-TOA, tangent is the appropriate ratio.

 

 

The angle that the rail line is inclined to the horizontal is approximately 2.6°.