# Light between 2 towers reflecting off a lake

1. Apr 20, 2017

### ArmChairPhysicist

• HW Template missing as it was moved from another forum
Please forgive me if I'm in breach of some etiquette with asking for help here, but calculus isn't my strongest subject, and I learn best by example.

I am currently studying in the Life Of Fred Calculus book (do I need to state copyright or something?). Im attempting to work a problem that reads as such:

"Two towers are 120 feet apart. The one on the left is 7 feet tall and the one on the right is 21 feet tall. A beam of light from the top of the left tower bounces off of a lake (note: I assume the lake spans the distance between the two towers, perfectly in line with them and not off somewhere else.) and hits the top of the other tower. Assuming the light takes the shortest path, how far from the base of the left tower will the beam strike the lake?" Authors note: "The algebra- not the calculus- is tough. After you have taken the derivative and set it equal to zero, try x= 10, 20, 30..... and one of those will work."

Things I know need to be done.
As the problem states, I'm looking for the length base of a triangle that has 7 feet (the tower) as its height and the beam of light as the hypotenuse. I know I'm going to need the almighty derivative of an equation relating to the triangle, but I'm not sure how to get that equation.
Is there anyone that has experience with this book, or calculus in general, that could help me?

Last edited by a moderator: Apr 20, 2017
2. Apr 20, 2017

### ArmChairPhysicist

Forgot one thing, I know the answer, the distance from the base is fifty feet. But I'm stuck on figuring out how to solve this problem to get that answer in the first place.

3. Apr 20, 2017

### phyzguy

Can you make a sketch of the situation? From this you should be able to write an expression for the length of the light path.

4. Apr 20, 2017

### ArmChairPhysicist

I can. Give me a minute to photograph it.

5. Apr 20, 2017

### Staff: Mentor

Hint: Assume the light reflects off of the lake at a distance X from the base of the left tower. Think of the distance traveled by the light as having two parts: Top of left tower to lake (point X) and lake to top of right tower. Each part is the hypotenuse of a triangle: use a bit of trig to express the total distance traveled as a function of X.

6. Apr 20, 2017

### ArmChairPhysicist

If I am understanding the problem, I need to find the equation of line BF, take the equation's derivative (praise the almighty derivative), set the derivative equal to zero, and then solve by algebra. Where I've gotten stuck is finding out how to get the equation of BF so that AF and DF connect to the two towers as described in the problem. I'm sure that I've forgotten something extremely simple, but I'm not seeing where to go from here.

7. Apr 20, 2017

### ArmChairPhysicist

Ah there it is, trig. I'll try this

8. Apr 20, 2017

### Staff: Mentor

If it's a reflection, doesn't that constrain the angles of incidence and reflection...?

9. Apr 20, 2017

### ArmChairPhysicist

I thought about the laws of reflection, but given the confines of the book (calculus and not physics) I doubted the author would assume the reader would know or remember those rules, considering they aren't mentioned

10. Apr 20, 2017

### Staff: Mentor

You're not supposed to use that fact. But that will be the answer.

11. Apr 20, 2017

### Staff: Mentor

Right. You are actually using Fermat's principle of least time, which can be used to derive the law of reflection. For this exercise they want you to use calculus.

12. Apr 20, 2017

### Staff: Mentor

Oopsies! (Kobayashi-Maru...)

13. Apr 20, 2017

### phyzguy

Your diagram is incomplete. Draw the point F on the surface of the lake (just guess where it is), then draw the lines BF and DF. You should then be able to write expressions for the length of the line BF in terms of x and the length of the line DF in terms of x.

14. Apr 20, 2017

### Staff: Mentor

Draw the point F on your diagram. (The distance BF is what I called "X".) Then draw lines AF and DF; those are the two parts of the light's path that I referred to.

Use trig to find expressions for lengths AF and DF.

15. Apr 20, 2017

### ArmChairPhysicist

I've done that, working on the trig now.

16. Apr 20, 2017

### ArmChairPhysicist

In triangle abf , the only known values are angle abf = 90' and side AB = 7 feet. I don't know the hypotenuse, because to find that I need side BF, which is the whole reading I'm working this, to find BF. And I must be overlooking a trig function, because to use any of the six, I need at least another angle or side.

17. Apr 20, 2017

### ArmChairPhysicist

And technically I already know BF= 50, but working the problem backwards isn't going to teach me much.

18. Apr 20, 2017

### phyzguy

You 'know' the side BF, it's distance is x. So write the hypotenuse AF in terms of AB (7 feet) and x.

19. Apr 20, 2017

### ArmChairPhysicist

But can I even solve that with two out of three variables being unknown?

20. Apr 20, 2017

### ArmChairPhysicist

7^2 + X^2 = C^2. C=AF