In searching the bottom of a

pool at night, a watchman
shines a narrow beam of light
from his flashlight, 1.3 m
above the water level, onto the
surface of the water at a point
2.7m from the edge of the
pool. Where does the spot of
light hit the bottom of the pool,
measured from the wall
beneath his foot, if the pool is
2.1m deep.

First use trig to determine the incidence angle of the beam hitting the water. That angle is

I = arctan 2.7/1.3 = arctan 2.077 = 64.3 degrees

The use Snell's law to calculate the angle of refraction, R, in the water. Assuming the refractive index of water is 1.33,
1.33 sin R = sin 64.3
sin R = 0.6775
R = 42.6 degrees
1.33 sin R

The beam goes an additional horizointal distance under the water, given by
X = 2.1 tan R
Add that to 2.7 m for the answer

Well, it seems like the watchman is in for a little poolside adventure! Let's see where that mischievous beam of light will hit the bottom of the pool.

First, we need to figure out the distance from the wall beneath his foot to where the beam of light hits the water. We can use some good old Pythagoras here. The distance from the edge of the pool to the wall beneath his foot is 2.7 m, and the distance from the flashlight to the edge of the pool is 1.3 m. So, we add those two distances together: 2.7 m + 1.3 m = 4 m.

Now, we know that the pool is 2.1 m deep, and the flashlight is 1.3 m above the water level. If we subtract those two distances, we get: 2.1 m - 1.3 m = 0.8 m.

Therefore, the spot of light will hit the bottom of the pool, measured from the wall beneath the watchman's foot, at a distance of 0.8 m.

Just remember, if you ever find yourself searching for treasure at the bottom of a pool, watch out for those sneaky beams of light!

To solve this problem, we can use the principles of similar triangles. Let's assume that the spot of light hits the bottom of the pool at a point x meters from the wall beneath the watchman's foot.

From the given information, we know the following lengths:
- The distance from the flashlight to the water surface is 1.3 m.
- The distance from the flashlight to the edge of the pool is 2.7 m.
- The depth of the pool is 2.1 m.

Now, let's set up a proportion using the similar triangles:

(1.3 m)/(2.7 m) = (x m + 2.1 m)/(x m)

Simplifying the proportion, we get:

1.3/2.7 = (x + 2.1)/x

Cross-multiplying, we have:

1.3x = 2.7(x + 2.1)

Expanding the equation:

1.3x = 2.7x + 5.67

Rearranging the equation:

2.7x - 1.3x = 5.67

1.4x = 5.67

Dividing both sides by 1.4:

x = 5.67/1.4

x ≈ 4.05

Therefore, the spot of light hits the bottom of the pool approximately 4.05 meters from the wall beneath the watchman's foot.

To find the spot of light where it hits the bottom of the pool, we need to analyze the light beam's path and calculate its point of intersection with the pool's bottom. Let's break down the problem into steps:

Step 1: Visualize the problem
Imagine a rectangular pool with a watchman standing at one of the edges. The pool has a depth of 2.1 m, and the watchman shines a narrow beam of light from his flashlight, positioned 1.3 m above the water surface.

Step 2: Determine the light beam's path
Since the beam of light is narrow, we can treat it as being parallel until it reaches the water's surface. From there, it will bend at the water-air boundary and continue along a new path. By applying the laws of reflection, we know that the angle of incidence is equal to the angle of reflection.

Step 3: Calculate the angle of incidence
To find the angle of incidence, we can use trigonometry. Since cosine is adjacent over hypotenuse, the angle of incidence can be calculated as follows:

cos(θ) = (distance from the edge of the pool)/(distance from the edge of the pool + distance from the wall beneath the watchman's foot)
cos(θ) = 2.7m/(2.7m + x), where x is the distance from the wall beneath the watchman's foot to the spot of light on the pool's bottom.

Step 4: Calculate the angle of reflection
Since the angle of incidence is equal to the angle of reflection, we can use the angle of incidence calculated in step 3.

Step 5: Calculate the distance from the wall beneath the watchman's foot to the spot of light on the pool's bottom
Using trigonometry again, we can find the distance (x) from the wall beneath the watchman's foot to the spot of light on the pool's bottom. We can use sine to relate the angle of reflection, the distance from the water's surface to the bottom of the pool, and the unknown distance (x).

sin(θ) = (distance from the water's surface to the bottom of the pool)/x
sin(θ) = 2.1m/x

Step 6: Solve for x
To find the value of x, we need to isolate it in the equation found in step 5:

x = (2.1m)/sin(θ)

Step 7: Substitute θ with the calculated angle of reflection from step 4 and solve for x.

Now that we have established the steps, we can proceed with the calculations to find the spot where the light hits the bottom of the pool.