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.