To solve this problem, we can use trigonometry. Specifically, we’ll use the sine function, as we are given the angle of elevation and the opposite side (14 ft), and we need to find the hypotenuse (the length of the ladder).

Given:

• Opposite side ($h$) = 14 ft
• Angle of elevation ($\theta$) = 71°
• We need to find the hypotenuse ($L$), which represents the ladder's length.

Solution:

The sine function is defined as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Rearranging to solve for the hypotenuse: $L = \frac{\text{opposite}}{\sin(\theta)}$

Plugging in the values: $L = \frac{14}{\sin(71^\circ)}$

Calculating this:

1. $\sin(71^\circ) \approx 0.9455$.
2. Therefore, $L = \frac{14}{0.9455} \approx 14.8$ ft.

Answer:

The distance from the bottom of the ladder to the side of the house is approximately 14.8 ft.

Would you like more details on this solution, or have any questions?

Here are some related questions for further exploration:

1. How would the solution change if the angle were different, say 60°?
2. What trigonometric function would we use if we were given the hypotenuse and needed to find the adjacent side?
3. How can we calculate the horizontal distance from the ladder’s base to the house?
4. What is the significance of rounding in real-world measurements?
5. How does changing the ladder's angle of elevation affect its stability?

Tip: In right triangle problems involving angles, determine which trigonometric function (sine, cosine, or tangent) to use based on the sides you have and need to find.