Heights And Distances

Let the height of the first lamp post be , and the height of the second lamp post be . The man divides the distance between the two posts in the ratio 1:2. The total distance between the posts is 30 m, so the distance between the man and the first lamp post is: The distance between the man and the second lamp post is: Now, we can apply similar triangles. The height of the first lamp post and the distance between the first lamp post and the man form a triangle, as does the height of the man and the distance between the man and the second lamp post. The ratio of the heights of the lamp post to the man is equal to the ratio of their respective distances. Solving for :

The correct option is (D):

Given, the length of the room m, breadth m, and height m.
The diagonal of the base of the room can be calculated using the Pythagoras theorem: Now, to find the angle of elevation , we consider the right triangle formed by the diagonal of the base, the height of the room, and the line of sight to the top corner. The angle of elevation is given by: Therefore,

The correct option is (C):

Correct answer: 30°

Explanation:
Let the height of the pole be meters and the length of the shadow be meters. In this case, the tangent of the angle of elevation is given by: We know that: Therefore, the angle of elevation is .

Hence, the angle of elevation of the sun is 30°.

Correct answer: 13

Explanation:
The height difference between the two poles is: The horizontal distance between the poles is: Now, use the Pythagorean Theorem to find the distance between the tops of the poles:

Hence, the distance between the tops of the poles is 13 meters.

To calculate the distance between the points and , we use the distance formula:

Here, and , so: Thus, the distance is .

Let the distance between the towers' feet be . We are given that the angle of elevation from the midpoint of the line joining their feet to the top of the first tower is , and to the second tower is . The height of the towers are and respectively.
The tangent of an angle in a right-angled triangle is given by the ratio of the opposite side to the adjacent side. Therefore, for the first tower: For the second tower: Now, the ratio of the heights is:

The correct option is (B):

Let the height of the tree be .

The tree breaks at some height from the ground.

The broken part touches the ground at a distance of 10 meters from the foot of the tree and makes an angle of with the ground.

Let the length of the broken part be . Then .

We have a right-angled triangle with base 10 meters, height , and hypotenuse .

The angle between the base and the hypotenuse is .

Then, we have:

The entire height of the tree is meters.

Let and be the heights of the two towers.

Let the distance between the feet of the towers be .

The midpoint of the line joining their feet is at a distance from each tower.

The angles subtended at the midpoint are and respectively.

Then, we have $ \frac{h_1}{h_2} h_1:h_2 1:3$.

Step 1: Use Let the horizontal distances of cars from the base be and . For angle : For angle :
Step 2: Total distance between cars:

Step 1: Use trigonometric relation.
We are given:
Distance from wall (adjacent side) = ft
Angle with ground =
We need the hypotenuse (ladder length)
From ,

Step 1: Understand the concept of line of sight
The line of sight is the straight line drawn from the eye of the observer to the object being observed.
Step 2: Determine direction of observation
The question says: "when it is above the horizontal level"
This means the observer is looking upward from the horizontal
Step 3: Identify the correct terminology
If the observer looks above the horizontal: the angle formed is called the angle of elevation
If the observer looks below the horizontal: the angle formed is called the angle of depression
Step 4: Eliminate incorrect options
(2) is angle of depression – not applicable here
(3) is right angle – not relevant
(4) is "none of these" – incorrect