A jewel thief decides to hide a stolen diamond by placing it at the bottom of a crystal-clear fountain. He places a circular piece of wood on the surface of the water and anchors it directly above the diamond at the bottom of the fountain, as shown below. If the fountain is x = 1.65 m deep, find the minimum diameter of the piece of wood that would prevent the diamond from being seen from outside the water.

The outer edge of the circular wood piece should be at a location such that rays from the diamond to the surface correspond to the critical angle for total internal reflection. Let R be the radius of the wood piece and =x the depth. The incidence angle of a ray striking the edge will be
arctan R/x. This must equal the critical angle, which is arcsin 1/N, where N is the index of refraction.

R/x = tan (critical angle)

The critical angle is arcsin 1/1.33 = 48.7 degrees. Its tangent is 1.14
Therefore R/x = 1.14
The diameter you want us 2R. You know that x = 1.65 m.

Rays that go beyond the edge of the wood disc will exceed the critical and will be totaly refkected at the water's surface. Therefore the diamond cannot be seen from above.

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The minimum diameter of the piece of wood is 2 x 1.65 m = 3.3 m.

Wow, it looks like you've added some random letters there! I guess even bots can have a little fun with gibberish sometimes. But let's get back to the question at hand. The minimum diameter of the piece of wood that would prevent the diamond from being seen from outside the water is determined by the angle of total internal reflection. To calculate this, we use the formula R/x = tan(critical angle). Since the critical angle is 48.7 degrees, its tangent is 1.14. Therefore, the ratio R/x should be 1.14. Since x is given as 1.65 m, we can calculate the minimum diameter (2R) using the values we have. No gibberish required!

To find the minimum diameter of the piece of wood that would prevent the diamond from being seen from outside the water, we'll use the formula R/x = tan(critical angle).

Given:
Depth of the fountain, x = 1.65 m
Index of refraction, N = 1.33

First, let's find the critical angle:
arcsin(1/N) = arcsin(1/1.33) ≈ 48.7°

Next, calculate the tangent of the critical angle:
tan(critical angle) = tan(48.7°) ≈ 1.14

Now, we can substitute these values into the equation:
R/x = 1.14

Since we're looking for the diameter, and we know that diameter = 2R, we can rearrange the equation to solve for R:
R = (1.14)(x)

Substituting x = 1.65 m:
R = (1.14)(1.65) ≈ 1.88

Finally, the diameter of the wood piece will be twice the radius:
Diameter = 2R = 2(1.88) = 3.76 m

Therefore, the minimum diameter of the wood piece that would prevent the diamond from being seen from outside the water is approximately 3.76 meters.

To find the minimum diameter of the piece of wood that would prevent the diamond from being seen from outside the water, we can start by using the equation R/x = tan(critical angle).

Given:
Depth of the fountain, x = 1.65 m
Index of refraction, N = 1.33
Critical angle = arcsin(1/N) = arcsin(1/1.33) = 48.7 degrees

Take the tangent of the critical angle: tan(critical angle) ≈ 1.14

Now, substitute the values into the equation:
R/x = tan(critical angle)
R/1.65 = 1.14

To find the minimum diameter, we need to find 2R, so multiply both sides of the equation by 1.65:
R = 1.14 * 1.65

Multiply the numbers:
R ≈ 1.881

Finally, to find the minimum diameter, we multiply R by 2:
2R ≈ 2 * 1.881 = 3.762

Therefore, the minimum diameter of the piece of wood that would prevent the diamond from being seen from outside the water is approximately 3.762 meters.