A surveyor walks 500m up a hill which slopes at an angle of 30$^o$. Calculate the vertical height through which he rises

1. Analyze the question

We are given a scenario involving a surveyor walking up a hill. We have:

• The distance walked along the slope (the hypotenuse of a right-angled triangle) = 500m
• The angle of inclination (the angle the slope makes with the horizontal) = 30°
• We need to find the vertical height (the side opposite the angle).

2. Explain the concept

This is a trigonometry problem. In a right-angled triangle, the relationship between the angle, the opposite side (height), and the hypotenuse (slope) is defined by the Sine function:

Where:

• $\theta = 30^\circ$
• Hypotenuse = 500m
• Opposite = Vertical height ($h$)

3. Show the solution

Using the formula:

We know from trigonometric ratios that $\sin(30^\circ) = 0.5$ (or $\frac{1}{2}$).

Substitute the value:

To find $h$, multiply both sides by 500:

4. Identify the answer

The vertical height is 250m. Comparing this to the given options: A. 252m B. 500m C. 250m D. 255m

The correct option is C.

5. Tip

Remember the mnemonic SOH CAH TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Whenever you see "vertical height" in a slope problem, you are almost always looking for the "Opposite" side, so SOH is your best friend!