Question

The surveyor sets up the instrument to determine the height at the highest point on the hill. She determines that she is standing at a horizontal distance of 100 m and sights the top of the hill using an angle of 17°

i) Let 'h' represent the height of the hill and 'd' represent the horizontal distance.
ii) Set up, the appropriate trigonometric ratio to determine the height of the hill. (DO NOT SOLVE ONLY SET UP)
iii) Replace the variables with the appropriate values.

Answer 1

i)
`\(h = d * \tan(\theta)\)`

ii)
`\(\tan(\theta) = (h)/(d)\)`

iii)
`\(h = 100 * \tan(17°)\)`

Step-by-step explanation:

To determine the height of the hill, we can utilize the tangent function in trigonometry, which relates the angle of elevation
`(\(\theta\))`to the height (\(h\)) and horizontal distance
`(\(d\)).` The trigonometric ratio for this scenario is given by
`\(\tan(\theta) = (h)/(d)\),` where
`\(h\)` represents the height of the hill and \(d\) is the horizontal distance from the surveyor to the base of the hill. Rearranging this formula gives us the equation h =
`d * \tan(\theta)\),`which allows us to find the height of the hill when given the angle and horizontal distance.

In this specific case, the horizontal distance
`(\(d\))` is 100 meters, and the angle of elevation (\(\theta\)) is 17 degrees. By substituting these values into the formula
`\(h = 100 * \tan(17°)\),` we can calculate the height of the hill. This computation results in the determination of the hill's height based on the surveyor's position and the angle at which the top of the hill is sighted.

Trigonometric ratios provide a powerful tool for solving problems involving angles and distances in various contexts, allowing us to calculate unknown values such as heights, distances, or angles in a triangle when certain information is given. The tangent function, in particular, helps determine the height of the hill in this scenario, enabling accurate measurements without needing to physically access the top of the hill.

"Trigonometry, through the tangent function, relates the angle of elevation to the height and horizontal distance, allowing precise calculations of otherwise inaccessible heights, as demonstrated in this hill measurement scenario."