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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

Milan uses an escalator at the railway station.

If he runs up 8 steps of the escalator, then it takes him 40.0 seconds to reach the top of the escalator.

If he runs up 15 steps of the escalator, then it takes him only 22.5 seconds to reach the top of the escalator.

How many seconds would it take Milan to reach the top, if he did not run up any steps of the escalator at all? Also, how many steps are visible, when the escalator is stationary?

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Why did the vector cross the road?

It wanted to be normal.

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

correct

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

(sinx)^4-(sinx)^5-(cosx)^5 = 2

solve for x

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi smiyc86;

Here is a quick and very ugly way to get the real roots of (sinx)^4-(sinx)^5-(cosx)^5 = 2

First I notice that big 2 on the rhs and guess that one of the extrema of sin(x) and cos(x) are the answer. We have:

Just trying the first one yields (-1)^4 - (-1)^5 - 0 which equals 2 so x = -π/2 is one solution. Since sin is periodic

with c=0,1,2,3... are all solutions.

There are also complex solutions but I could not get them without resorting to an iterative method so I left them out.

bobbym

*Last edited by bobbym (2009-04-17 00:06:33)*

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

hi bobbym,

very good observation

and in fact your answer is correct ...

I would give a slightly better way

rewrite the equn as

(sin x)^4 - 2 = (sin x)^5 + (cos x)^5

now the max value of LHS is -1

and the min value of RHS = -1

thus if the equation holds true ... both the sides are got to be equal to -1

hence u can come to the solution ...

*Last edited by smiyc86 (2009-04-21 20:22:33)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Thanks for providing me with another way to do it.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

# 3

Find the smallest number N which has the following properties:

its decimal representation has 6 as the last digit.

If the last digit 6 is erased and placed in front of the remaining digits, the resulting number is four times as great as the original number N.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi smiyc86;

solution to #3

solution

*Last edited by bobbym (2009-04-22 03:53:16)*

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**smiyc86****Member**- Registered: 2009-03-19
- Posts: 78

nice ....

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