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

1. Substitute the given values into the cosine formula:

   \(\cos(45^\circ) = \frac{15 \text{ m}}{\text{Hypotenuse}}\)

2. Recall the value of \(\cos(45^\circ)\):

   \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)

3. Set up the equation:

   \(\frac{1}{\sqrt{2}} = \frac{15 \text{ m}}{\text{Hypotenuse}}\)

4. 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.