Problem 65 - Algebra and Trigonometry

Question

Describe a situation in which a right triangle and a trigonometric function are used to measure a height or distance that would otherwise be inconvenient or impossible to measure.

Step-by-Step Solution

Answer

The height of the lighthouse can be measured using the distance the ship moved parallel to the shore and the angle of elevation of the lighthouse from the ship, using the formula: Height = d * tan(\(\theta\) ) where d is the distance moved by the ship and \(\theta\) is the angle of elevation.

Step 1: Describe the Scenario

Consider a scenario where there's a lighthouse on the seashore and a ship at sea, wanting to calculate the height of the lighthouse. It would be inconvenient or in some cases impossible to measure this height directly. So, trigonometry can be used here.

Step 2: Detail the Process

The ship will move a certain distance parallel to the shoreline, let's say 'd', forming a right triangle with the lighthouse and its initial and final position. The sailors on the ship measure the angle of elevation of the lighthouse from the ship at its final position, let's say this angle is '\(\theta\)'. Now, using the trigonometric function 'tan', which is opposite side/adjacent side, in this case, the height of the lighthouse over the distance 'd'.

Step 3: Formulate the Calculation

The height of the lighthouse can then be calculated using the formula: Height = d * tan(\(\theta\) ).