Ladder Length Calculation: Angle of Elevation Problem
This problem involves finding the length of a ladder leaning against a wall using trigonometry. We can model this situation as a right-angled triangle where:
- The ladder represents the hypotenuse (L).
- The wall represents the perpendicular side.
- The distance from the foot of the ladder to the wall is the adjacent side.
Applying Trigonometry
We are given:
- Angle of elevation ($\theta$) = $45^\circ$
- Distance from the wall (Adjacent side) = $4\sqrt{2}$ m
To find the length of the ladder (Hypotenuse), we can use the cosine function, which relates the adjacent side and the hypotenuse to the angle:
$ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $
Calculating Ladder Length
Substitute the known values into the formula:
$ \cos(45^\circ) = \frac{4\sqrt{2}}{L} $
We know that $\cos(45^\circ) = \frac{1}{\sqrt{2}}$. Therefore:
$ \frac{1}{\sqrt{2}} = \frac{4\sqrt{2}}{L} $
Now, solve for L:
$ L = 4\sqrt{2} \times \sqrt{2} $
$ L = 4 \times (\sqrt{2})^2 $
$ L = 4 \times 2 $
$ L = 8 $
Final Answer
The length of the ladder is 8 metres.
Similar Questions
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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?
