We can solve this problem using trigonometry, specifically the sine function.

Given:

• The ladder makes an angle of $30^\circ$ with the wall.
• The height of the wall (which is the opposite side of the angle) is $8$ meters.

We use the sine formula: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Here:

• $\theta = 30^\circ$
• Opposite side = $8$ meters
• Hypotenuse = Length of the ladder, let's call it $L$

Rearranging the formula to solve for $L$: $L = \frac{\text{opposite}}{\sin(\theta)} = \frac{8}{\sin(30^\circ)}$

Since $\sin(30^\circ) = 0.5$, we substitute: $L = \frac{8}{0.5} = 16 \text{ meters}$

Answer

The length of the ladder is 16 meters.

Would you like a detailed explanation or have any questions?

Here are five related questions for further exploration:

1. What is the length of the ladder if the angle changes to 45 degrees with the same wall height?
2. How does the ladder's length change if the wall's height increases to 10 meters while keeping the angle at 30 degrees?
3. What would be the angle of the ladder if its length is 20 meters, and it reaches a wall height of 8 meters?
4. How can you calculate the distance between the foot of the ladder and the wall?
5. What is the significance of using sine, cosine, and tangent in solving ladder problems?

Tip: In right-angled triangle problems, always identify which side is opposite, adjacent, or the hypotenuse relative to the angle. This helps in selecting the correct trigonometric function.