The smallest angle flashlight beam can make with horizontal

AI Thread Summary
To determine the smallest angle a flashlight beam can make with the horizontal to illuminate the bottom of a cylindrical tank, the light must hit the bottom directly without reflecting off the sides. The tank's dimensions are 10.5 meters in depth and 9.45 meters in diameter, leading to a necessary calculation involving the tangent function. The angle is calculated using the formula tan(θ) = y/x, where y is the depth and x is the radius of the tank. The discussion emphasizes that the light beam should ideally strike the bottom at a point that allows for the lowest angle possible, without any reflections complicating the scenario. Understanding the geometry of the tank is crucial for solving the problem accurately.
armxneo
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Homework Statement


At the aquarium where you work, a fish has gone missing in a 10.5m -deep, 9.45m -diameter cylindrical tank. You shine a flashlight in from the top edge of the tank, hoping to see if the missing fish is on the bottom. What’s the smallest angle your flashlight beam can make with the horizontal if it’s to illuminate the bottom?

nwater=1.33 (Not sure if this is relevant)

Homework Equations



tan(θ)=y/x
n1sinΘ=n2cosθ

The Attempt at a Solution



I first drew a picture of a cylinder with height 10.5 meters and 9.45 meters
I assume that in order to illuminate the bottom of the tank, the light needs to hit the side of the tank at half of it's height (if this is incorrect, then I am not sure how to determine where the light hits the side of the tank)

At height 10.5/2= 5.25 meters, I solved for Θ; tan(Θ)=5.25/9.45
tan-1(5.25/9.45)= 29.05 degrees

I used this equation in snell's law:
1.33sin(29.05)=(1)sin(θair)
θ=40.23.
I found the θ to the normal, so to find θ to the horizontal: 90-40.23= 49.77 degrees.

This answer is incorrect, so any help would be appreciated!
 
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Why at half its height? Are you assuming reflection off the side? I would not assume that.
 
I know it reflects off the side because this would give a smaller angle to the horizontal (which is what the question is asking me to optimize for). I am unsure about how I would find an angle, and where the light would reflect off of if it was not at half the height.
 
armxneo said:
I know it reflects off the side because this would give a smaller angle to the horizontal (which is what the question is asking me to optimize for).
That does not follow. Suppose it does not reflect. It is possible to hold the light so that it illuminates (some part of) the bottom; at a vey low angle it will not iluminate the bottom; therefore there is a least angle at which it illuminates the bottom.
Conversely, if it can reflect off one side, why not both sides?
 
Ok. Based on that I have no idea how to approach the problem with the given information.
 
armxneo said:
Ok. Based on that I have no idea how to approach the problem with the given information.
This makes me think you have misunderstood the set up, but I can't guess in what way. Draw a cross section of the tank. It's a rectangle. The source of light is at top left (say). To just reach the bottom, without reflections, at the lowest possible angle, where will the light beam strike the bottom?
 
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