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Interesting Concepts...

Hello, I have a few questions that I have made myself, but I don't know how to answer them.

1) Say that there is an isosceles triangle ABC. Two of the equal angles are 1/∞. What is the other angle?

2) Is a sphere really a sphere if you can draw a tangent to it?

3) Assume that there exists a perfect sphere, that can have no tangent drawn to it. If one were to drop the sphere, what would happen to it? It couldn't touch the ground, because it can't have a tangent drawn to it.

4) The graph of tan(x) has several vertical asymptotes on it. But what is the width of this asymptote? Moreover, what is the distance between the edge of the curve on the left-hand side of the asymptote and the edge of the curve on the right-hand side (at the centre)?

5)

Shown above is the formula for the Fibonacci sequence. But how is this derived?

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I'll have more questions soon, but these ideas have been boggling my mind for a while now. Can anyone help me out?

Devantè

Real Member

Registered: 2006-07-14

Posts: 6,400

Re: Interesting Concepts...

Hello, I have a few questions that I have made myself, but I don't know how to answer them.

1) Say that there is an isosceles triangle ABC. Two of the equal angles are 1/∞. What is the other angle?

Since the sum of angles of a triangle is 180 degrees, or

radians, I would say the answer is

, or rather,

while the sum of the other angles,

goes to 0. However, I'd assume that, if we say that, if the two equal angles approach 0, then we'd simply get a straight line.

2) Is a sphere really a sphere if you can draw a tangent to it?

Why do you think a sphere can't have tangent lines? At a given point, a point on the surface of a sphere has infinitely many tangent lines, which make up the tangent plane at that point.

4) The graph of tan(x) has several vertical asymptotes on it. But what is the width of this asymptote? Moreover, what is the distance between the edge of the curve on the left-hand side of the asymptote and the edge of the curve on the right-hand side (at the centre)?

An asymptote is a line, so it has the width of a line, which I guess is 0. Since tan(x) has asymptotes at the points

, the second part sounds like you're asking about the distance between the values

for

.

5)

Shown above is the formula for the Fibonacci sequence. But how is this derived?

The Fibonacci sequence is defined by F[sub]0[/sub]= 1, F[sub]1[/sub]= 1,  F[sub]n+2[/sub]= F[sub]n+1[/sub]+ F[sub]n[/sub].

   A standard method of trying to solve such recursion formulas is to try something of the form F[sub]n[/sub]= a[sup]n[/sup].  Then, of course, F[sub]n+1[/sub]= a[sup]n+1[/sup] and F[sub]n+2[/sub]= a[sup]n+2[/sup] so the equation becomes a[sup]n+2[/sup]= a[sup]n+1[/sup]+ a[sup]n[/sup].  If we divide the entire equation by a[sup]n[/sup] we arrive at a[sup]2[/sup]= a+ 1 or the quadratic equation a[sup]2[/sup]- a- 1= 0.  Solving that by the quadratic formula,

That tells us that

and

satisfy that equation.  Since this is a linear equation, any solution must be of the form

for some numbers A and B.
In particular, if n= 0

and, if n= 1

  Solving those two equations for A and B, we get

and

Putting all of that together, we have

Last edited by Devantè (2009-12-21 06:05:13)