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?
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Important Questions from Heights and Distances
Two pillars are placed vertically 8 feet apart. The height difference of the two pillars is 6 feet. The two ends of a rope of length 15 feet are tied to the tips of the two pillars. The portion of the length of the taller pillar that can be brought in contact with the rope without detaching the rope from the pillars is
- 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?
