$15\sqrt{2}\text{m}$
Finding Ladder Length Using Trigonometry
This problem involves finding the length of a ladder leaning against a wall using basic trigonometry.
Understanding the Setup
We can model this situation as a right-angled triangle where:
- The ladder represents the hypotenuse.
- The distance from the foot of the ladder to the wall is the adjacent side (to the angle of elevation).
- The height the ladder reaches on the wall is the opposite side.
We are given:
- Angle of elevation (\(\theta\)) = \(45^\circ\)
- Distance from the wall (Adjacent side) = 15 m
- We need to find the length of the ladder (Hypotenuse).
Applying Trigonometric Ratios
The trigonometric ratio that relates the adjacent side and the hypotenuse is the cosine function:
\(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
Calculation Steps
- Substitute the given values into the cosine formula:
\(\cos(45^\circ) = \frac{15 \text{ m}}{\text{Hypotenuse}}\)
- Recall the value of \(\cos(45^\circ)\):
\(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)
- Set up the equation:
\(\frac{1}{\sqrt{2}} = \frac{15 \text{ m}}{\text{Hypotenuse}}\)
- Solve for the Hypotenuse (length of the ladder):
\(\text{Hypotenuse} = 15 \text{ m} \times \sqrt{2}\)
\(\text{Hypotenuse} = 15\sqrt{2} \text{ m}\)
Conclusion
The length of the ladder is \(15\sqrt{2}\) m.
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 the sun, when the length of the shadow of a tree is equal to the height of the tree, 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?
