Question

A ladder 7 m long makes an angle of 30\(^\circ\) with the wall. Find the height of the point on the wall where the ladder touches the wall.

Hint:

Always draw a diagram for problems involving heights and distances. Carefully label the sides and angles. Pay close attention to whether the angle is an angle of elevation (with the horizontal) or an angle given with a vertical object.

Answer 1

Step 1: Understanding the Concept:
This problem involves a right-angled triangle formed by the ladder, the wall, and the ground. We need to use trigonometric ratios to find the height on the wall. It is crucial to correctly identify which angle is given.

Step 2: Key Formula or Approach:
Let's visualize the setup. The wall is vertical, the ground is horizontal, and the ladder leans against the wall, forming the hypotenuse. Let \(h\) be the height on the wall. The length of the ladder is the hypotenuse = 7 m. The angle given is between the ladder and the wall, which is \(30^\circ\). In the right-angled triangle, the height \(h\) is the side adjacent to the \(30^\circ\) angle. The appropriate trigonometric ratio is cosine: \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).

Step 3: Detailed Explanation:
Using the cosine ratio: \[ \cos(30^\circ) = \frac{h}{7} \] We know the standard value \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\). \[ \frac{\sqrt{3}}{2} = \frac{h}{7} \] Solve for \(h\): \[ h = 7 \times \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2} \text{ m} \] Alternative Method: The angle between the ladder and the ground would be \(90^\circ - 30^\circ = 60^\circ\). In this case, the height \(h\) is the side opposite to the \(60^\circ\) angle. Using the sine ratio: \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\). \[ \sin(60^\circ) = \frac{h}{7} \] We know \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). \[ \frac{\sqrt{3}}{2} = \frac{h}{7} \implies h = \frac{7\sqrt{3}}{2} \text{ m} \] Both methods yield the same result.

Step 4: Final Answer:
The height of the point on the wall is \(\frac{7\sqrt{3}}{2}\) m.