A spotlight on a boat is y = 2.6 m above the water, and the light strikes the water at a point that is x = 8.1 m horizontally displaced from the spotlight. The depth of the water is 4.0 m. Determine the distance d, which locates the point where the light strikes the bottom.
To determine the distance d, we can use the concept of similar triangles. Let's analyze the problem step by step.
The given information:
- The height of the spotlight above the water, y = 2.6 m.
- The horizontal displacement of the light from the spotlight, x = 8.1 m.
- The depth of the water, h = 4.0 m.
We can create two similar triangles in this scenario: one formed by the spotlight, the point on the water's surface where the light strikes, and the point where the light strikes the bottom of the water, and the other formed by the spotlight, the point where the light strikes the bottom, and the bottom of the water.
Now let's designate some variables to work with:
- Let d be the distance between the point where the light strikes the water's surface and the point where the light strikes the bottom.
- Let h' be the height of the point where the light strikes the bottom above the bottom of the water.
Since the two triangles are similar, we can set up a proportion between their corresponding sides:
(y + h') / d = y / x
Plugging in the given values, we get:
(2.6 + h') / d = 2.6 / 8.1
Now we can solve for d. Cross-multiplying the equation, we have:
8.1 × (2.6 + h') = 2.6 × d
20.66 + 8.1h' = 2.6d
Next, we know that the total height from the water's surface to the bottom is h + h', which is equal to 4.0 m. Thus, we can substitute h' with 4.0 - h in the above equation:
20.66 + 8.1(4.0 - h) = 2.6d
20.66 + 32.4 - 8.1h = 2.6d
52.06 - 8.1h = 2.6d
Finally, we can rearrange the equation to solve for d:
d = (52.06 - 8.1h) / 2.6
Now we can substitute the value of h (depth of the water) to find the distance d.