You are Here: Home >< Maths

# The Proof is Trivial! Watch

1. (Original post by k9markiii)
I liked that middle question can't be bothered with the last one.
For question 362, I made an edit.

Posted from TSR Mobile
2. (Original post by james22)
That's the one I meant, I'm curious what the solution is because I could not find any way to make simpler. I managed to convert it to several different integrals but they didn't look any nicer.
Well that's pretty much what I did, though it seemed satisfying so I'll type it up anyway:

(Original post by Lord of the Flies)
Problem 260***

Let so that

Taking the Laplace transform, applying Fubini's theorem and then using the standard result (proven by IBP or the exponential representation of sine) for the Laplace transform of , we have

Let so that

The starter question on an old USAMO paper:

Problem 366*

Find the average of the numbers ()
3. I have missed this integral.

Solution 260

Let me use the notation of Jkn's post.
.
Note that .
Hence, we can let and obtain
.
The last integral is an easy exercise in complex analysis.
What's this supposed to be? (its's a typo!)
You missed out the s. Typo?
5. (Original post by Arieisit)
Problem 362*

This is actually very similar to an A level question but without the added guidance that they usually provide.

A sports association is planning to construct a running track in the shape of a rectangle surmounted by a semicircle. Let the letter represents the length of the rectangular section and represents the radius of the semicircle.
Determine the length, , that maximises the area enclosed by the track.

EDIT

The perimeter of the track must be 600 metres.

I completely forgot to put that in. Sorry

600m running track hmm OK different from the 400m standard, nice. Well and where P is 600m. There are a number of ways to do this but it would be neatest with lagrange multipliers I think.

This implies the maximum area is enclosed when x = 0.

This makes sense as a circle contains the most area for a given perimeter of any shape.
Certainly didn't go for a straightforward method but using langrange multipliers is useful when you are adding constraints to a system. It seems an odd answer are you sure you've got the question right?
6. (Original post by Jkn)
What's this supposed to be? (its's a typo!)

You missed out the s. Typo?
Indeed. What I did is .
Indeed. What I did is .
Wow, that's brilliant! Beautiful solution!
Spoiler:
Show
Sorry to be an anus but the first expression is still typed wrong! And the half shouldn't come in to the second line (after adding it with the modified expression)

Btw, I'm currently fiddling around trying to see if I can find a closed form for and have come to a halt at which, despite being able to evaluate very neatly, I am yet to find a general expression for it (it decays into a mess of polygammas). I've also realised that, if this could be evaluated, we would have a taylor series for the gamma function. Is there a taylor series for the gamma function or am I going into the realm of the unknown?
8. (Original post by k9markiii)
Certainly didn't go for a straightforward method but using langrange multipliers is useful when you are adding constraints to a system. It seems an odd answer are you sure you've got the question right?
Yes, I am sure this time. My solution is much simpler but I agree, the answer is definitely odd.
9. (Original post by Arieisit)
Yes, I am sure this time. My solution is much simpler but I agree, the answer is definitely odd.
I was practising out some fun maths. I know you can do it through simultaneous equations but I was practising some maths I learnt last year to get ready for lectures starting next week.
10. (Original post by Arieisit)
Problem 362*

This is actually very similar to an A level question but without the added guidance that they usually provide.

A sports association is planning to construct a running track in the shape of a rectangle surmounted by a semicircle. Let the letter represents the length of the rectangular section and represents the radius of the semicircle.
Determine the length, , that maximises the area enclosed by the track.

EDIT

The perimeter of the track must be 600 metres.

I completely forgot to put that in. Sorry
You don't need to assign a numerical P here, so long as P is constant.

I suppose you have marked this as * so maybe you just want to show something nice - but surely the answer is trivial when you consider what a circle actually is and what some of its inherent properties are.
11. (Original post by Llewellyn)
You don't need to assign a numerical P here, so long as P is constant.

I suppose you have marked this as * so maybe you just want to show something nice - but surely the answer is trivial when you consider what a circle actually is and what some of its inherent properties are.
I think you quoted the wrong person. Lol

EDIT

Posted from TSR Mobile
12. (Original post by Arieisit)
Problem 362*

This is actually very similar to an A level question but without the added guidance that they usually provide.

A sports association is planning to construct a running track in the shape of a rectangle surmounted by a semicircle. Let the letter represents the length of the rectangular section and represents the radius of the semicircle.
Determine the length, , that maximises the area enclosed by the track.

EDIT

The perimeter of the track must be 600 metres.
Ignore this: I misread the question and maximised the area for the rectangle surmounted by a semi circle on either side
.
Wrong solution
Solution 362

Let A be the area of the track and let P be its perimeter. That means that:

To find the length of x that maximises the area enclosed by the track, it'd be helpful to find the area in terms of x alone.

The maximum value of the area occurs when the first derivative is equal to 0 and the second derivative is less than 0.

The area is maximised when the length is equal to 0 and the track is circular.
13. (Original post by Khallil)
Solution 362

Let A be the area of the track and let P be its perimeter. That means that:

To find the length of x that maximises the area enclosed by the track, it'd be helpful to find the area in terms of x alone.

The maximum value of the area occurs when the first derivative is equal to 0 and the second derivative is less than 0.

The area is maximised when the length is equal to 0 and the track is circular.
I have a slight qualm - your equations are for a circle when it is quite specifically a semi-circle that is mentioned.
14. (Original post by Arieisit)
Problem 362*
(Original post by DJMayes)
I have a slight qualm - your equations are for a circle when it is quite specifically a semi-circle that is mentioned.
Damn it! I read the question but my brain told me that the rectangle is surmounted by a semi circle on either side, like the race tracks in the Olympics

I hope this is right:

Solution 362

Let A be the area of the track and let P be its perimeter. That means that:

To find the length of x that maximises the area enclosed by the track, it'd be helpful to find the area in terms of x alone.

The maximum value of the area occurs when the first derivative is equal to 0 and the second derivative is less than 0.

The area is maximised when the length is equal to
15. Since nobody attempted Problem 364 I'll provide a hint or two.

Problem 364***

Determine the extreme values of

=

where = , = , and

Spoiler:
Show
Find all the intrinsic critical points of on M
Calculate .
Investigate whether or not attains the value at some point of M

Problem 367**

If = find

Problem 368**

Given that = and = prove that
16. (Original post by Arieisit)

Problem 368**

Given that = and = prove that

But note

So

So

So
17. Was my response to problem 362 correct?

(Original post by Arieisit)
Problem 367**

If = find
Solution 367

18. [QUOTE=Khallil;44692928]Was my response to problem 362 correct?

I got a similar answer if that helps

Posted from TSR Mobile
19. (Original post by Arieisit)
(Original post by Khallil)
Was my response to problem 362 correct?
Indeed it isn't

(Original post by Arieisit)
I got a similar answer if that helps
How similar?
20. (Original post by Khallil)
How similar?
Lets just say that yours looks more elegant.

Posted from TSR Mobile

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: December 11, 2017
Today on TSR

### What is the latest you've left an assignment

And actually passed?

### Simply having a wonderful Christmas time...

Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

## Groups associated with this forum:

View associated groups
Discussions on TSR

• Latest
• ## See more of what you like on The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

• The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.