From point 'P' on the ground, a long metallic cable is laid to the top of a cliff, making an angle of 30° with the ground such that the length of the cable laid is 450√3 metres. Find the distance between point 'P' and the foot of the cliff.
675 metres
The cable, the height of the cliff and the ground distance form a right triangle. The cable is the hypotenuse, the ground distance is the side adjacent to the \(30^\circ\) angle at P.
The relation between the adjacent side and the hypotenuse is \(\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}\), so \(\text{ground distance} = \text{cable} \times \cos\theta\).
With cable \(= 450\sqrt{3}\text{ m}\) and \(\cos 30^\circ = \dfrac{\sqrt{3}}{2}\): \(\text{distance} = 450\sqrt{3} \times \dfrac{\sqrt{3}}{2}\).
Since \(\sqrt{3} \times \sqrt{3} = 3\): \(\text{distance} = \dfrac{450 \times 3}{2} = \dfrac{1350}{2}\).
This evaluates to \(675\text{ metres}\).
Key concept: choose the trig ratio linking the known side to the unknown — here hypotenuse and adjacent, so cosine. Hence the distance between P and the foot of the cliff is 675 metres.
Similar Questions
A kite flying at a height of 60 metres above the ground is attached to string that makes an angle of 45° with the ground. What is the length (in metres) of the string?
A ladder rests against a vertical wall and makes an angle \(\theta\) with the horizontal ground such that \(\tan\theta = \dfrac{5}{12}\). If the length of the ladder is 13 meters, find the height at which the ladder touches the wall.
Important Questions from Heights and Distances
A peacock sitting at the top of a 3 meter high pole saw a snake approaching towards pole at a distance three times of the height of the pole. Then it jumping from pole will catch the snake at what distance from the pole if both are running with same speed ?
The foot of a ladder 25 m long is 7 m from the base of the building. If the top of the ladder slips by 4 m, then by how much distance will the foot of the ladder slide?
Two hotels stand 25 m apart. One of them is 70 m high and the angle of depression of the top of other as observed from the top of this hotel is 45°. Height of the other hotel is:
If the angles of elevation of a balloon from two consecutive kilometer-stones along a straight road are 30° and 60° respectively, then the height of the balloon above the ground will be:
The angle of elevation of the top of a tower from a point 20 m away from its base is 45 °. What is the height of the tower?
