A spotlight on a boat is y = 2.7 m above the water, and the light strikes the water at a point that is x = 8.6 m horizontally displaced from the spotlight (see the drawing). 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, which locates the point where the light strikes the bottom, we can use the concept of similar triangles. Here's how we can do it:
1. Draw a diagram to visually represent the situation. Based on the provided information, draw a right-angled triangle, with one side representing the height of the spotlight above the water (y) and another side representing the horizontal displacement of the light from the spotlight (x). Draw a third side representing the depth of the water (4.0 m).
2. Identify the similar triangles: In the diagram, we have two similar triangles. The first triangle is formed by the spotlight, the point on the water where the light strikes, and the point on the bottom where the light strikes (let's call this point B). The second triangle is formed by the spotlight, the point on the water where the light strikes, and the bottom of the pool.
3. Set up a proportion: We know that the height of the spotlight above the water (y) is 2.7 m and the horizontal displacement (x) is 8.6 m. We need to find the distance d, which represents the point where the light strikes the bottom. So, we can set up the following proportion between the sides:
y / x = (y + 4.0) / d
Substitute the given values:
2.7 / 8.6 = (2.7 + 4.0) / d
4. Solve the proportion for d: Cross-multiply the equation to get rid of the fractions:
2.7d = 8.6 * 6.7
d = (8.6 * 6.7) / 2.7
Calculate the value of d using a calculator:
d ≈ 21.03 m
So, the distance d, which locates the point where the light strikes the bottom, is approximately 21.03 meters.