To solve for the distance of the stone from the foot of the building, we can use trigonometry. Specifically, we'll use the tangent function, which relates the angles and sides of a right triangle.
Here's how we can do it step by step:
1. Understand the situation:
- We have a building height (opposite side) of 10 meters.
- The angle of depression to the stone is 69 degrees. Remember, the angle of depression from the top of the building to the stone on the ground is the same as the angle of elevation from the stone to the top of the building.
2. Identify the trigonometric relationship:
- We use the tangent function because it relates the angle of a right triangle with the length of the opposite side and the adjacent side:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
- Here, [tex]\(\theta\)[/tex] is 69 degrees, the opposite side is 10 meters, and the adjacent side is the distance from the foot of the building to the stone (which we need to find).
3. Set up the equation:
- [tex]\[ \tan(69^\circ) = \frac{10}{x} \][/tex]
- Here, [tex]\( x \)[/tex] is the distance from the foot of the building to the stone.
4. Solve for [tex]\( x \)[/tex]:
- Rearrange the equation to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{10}{\tan(69^\circ)} \][/tex]
5. Calculate the tangent of 69 degrees:
- Use a calculator to find [tex]\(\tan(69^\circ)\)[/tex]. Make sure your calculator is set to degrees.
[tex]\[ \tan(69^\circ) \approx 2.605 \][/tex]
6. Perform the division:
- Substitute [tex]\(\tan(69^\circ) = 2.605\)[/tex] into the equation:
[tex]\[ x = \frac{10}{2.605} \][/tex]
[tex]\[ x \approx 3.84 \][/tex]
7. Round the result to one decimal place:
- The distance from the foot of the building to the stone, rounded to one decimal place, is approximately 3.8 meters.
Therefore, the distance of the stone from the foot of the building is 3.8 meters.
