Calculating Sun's Angle of Elevation
This problem involves finding the angle of elevation of the sun under a specific condition using trigonometry.
Let H be the height of the tree and S be the length of its shadow.
Let \(\theta\) be the angle of elevation of the sun.
We can model this situation using a right-angled triangle:
- The height of the tree (H) is the side opposite to the angle \(\theta\).
- The length of the shadow (S) is the side adjacent to the angle \(\theta\).
The tangent function relates these values:
\(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{H}{S}\)
Applying the Given Condition
The question states that the length of the shadow is equal to the height of the tree:
\(H = S\)
Substitute this condition into the tangent formula:
\(\tan(\theta) = \frac{H}{H}\)
\(\tan(\theta) = 1\)
Finding the Angle
To find the angle \(\theta\), we need to determine the angle whose tangent is 1.
\(\theta = \arctan(1)\)
The angle whose tangent is 1 is \(45^\circ\).
\(\theta = 45^\circ\)
Therefore, the angle of elevation of the sun is \(45^\circ\).
Similar Questions
- A 1.8 m tall boy is standing at some distance from a 12 m tall building. The angle of elevation from his eyes to the top of the building increases from \(45^\circ\) to \(60^\circ\) as he walks towards the building. Find the distance (in meters) he walked towards the building.
- A tree breaks due to storm and the broken part bends, so that the top of the tree touches the ground making an angle of \(30^\circ\) with the ground. The distance between the foot of the tree to the point where the top touches the ground is 18 m. find the height of the tree (in metres).
- A 165cm tall man standing 500cm away from the wall looks at a hook on the wall with an angle of elevation of $45^\circ$. What is the height at which the hook is in the wall?
- The shadow of a candle fell on the table and made a right angled triangle with an angle of elevation of $60^\circ$ from the surface of the table to the upper tip of the flame of the candle. If the length of the candle is 21 cm, how long is the tip of the shadow on the table away from the base of the candle?
- In a right angled triangle, right-angled at $A$, the angle formed at $B$ is $45^\circ$, if the length of side $AB$ is 3m, then what is the length of the side $BC$?
There is a 10 m tall tower between two parallel roads. The angles of depression of the roads from the top of the tower are 30° and 45° respectively. Approximately, how far are the roads from each other?
- The angle of elevation of a ladder leaning against a wall is \(45^\circ\) and the foot of the ladder is 15m away from the wall. The length of the ladder is:
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?
