Math Is Fun Forum

They were √5 / 2 and √2.
1.118033988749894848204586834365638117720309179805762862135448...
and
1.414213562373095048801688724209698078569671875376948073176679...

This was a fun problem (constructing and solving simultaneous equations for a and r)

Hello!

The equations are also true for 2, 0, -2.

I am back to it.
Notice that bob's net is properly to scale, magnified to 4x. It makes the mathematics easy to visualise

The equation for the distance CY where Y is a point on PQ is:
CY = sqrt(y^2 + 1/4) where y = YQ/PQ

You can use the cosine rule to create a formula for any XY!

So XY = √(x^2 - √2*x(1-y) + (1-y)^2) where, again, x = PX/PR and X is any point on PR.

To prove the ant's path, once and for all, we just have to minimise AX + XY + CY. Which is this formula: √(2x^2 + 1/4) + √(x^2 - √2*x(1-y) + (1-y)^2) + √(y^2 + 1/4)

It looks like I was wrong about the ant. There is a solution to that equation which gives a marginally smaller distance than AP + PC ~ 1.6180.

I don't have the exact answers, but the distance is approximately 1.60759 if the ant walks from A to a point roughly 0.0939912 from P along PR, and then from there to a point roughly 0.128874 from P along PQ, and from there to C. That is, x is about 0.0664618 and y is about 0.871126 (where these are fractions along the lines PR from P and PQ from Q respectively).
This doesn't look to be quite the line drawn in bob's net to me (it looks like the climb up the wall is a little bit steeper), but all the equations are given to analyse it and why it is the way it is.

The termite proof will use a right triangle with small sides CY and XY and hypotenuse CX, where x = 1 - y (that equation ensures that the points referred to are in the same vertical line). I will continue this at some point, but that is simple enough now; it is not even a three-dimensional problem anymore

Agreed with bobbym (:

Oh, I should have paid more attention to the wording (: five MORE books, not five books. I agree with your solution, although the negative one seems counterintuitive (how is it that if they pay you $15 for every book, you could buy 5 more for $300 if they gave you $20 instead, or something? haha)

Every movie lover knows Casablanca!

Hello;

I agree with bobbym

-2n + 11 = -3n + 18

That's a pretty good reason (: Much worse ways to spend free time, and it is free for a reason.
But just to adjust for waking hours, it's really once every 23 minutes 38 seconds if you average 8 hours sleep. Just in case you were wondering P:

bobbym, how do you have so many posts? You've averaged 40.6 posts per day for the last six and a half years as of today. One every 35 and a half minutes. For six and a half years. It's obscene. lol