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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

Long time no see. I am 13 now and am in my second year of university. As you perhaps may not recall, my extremely partial Math teacher has thankfully retired and in my current course work I am facing some tribulations. Apparently my teacher nowadays gets excessively challenging Calculus questions from the Internet.

There is a new population of deer in a field in a National Park. Their growth can be modeled logistically (not exponentially - after all, there is only so much food and living space) based on the formula (deer vs. time): d(t)=(1500et-20)/(1000+20et-20). What is that field's carrying capacity? (The maximum number of deer it can sustain?)

I think it is 75 because of Hopital rule.

Annie has a 2-inch piece of chalk. Each inch of chalk can draw 1000 feet of lines. She wants to draw a curve modeled by the equation x3/2. She envisions the curve drawn on a plot where each unit on the x-axis and y-axis is equal to 1 yard (1 yard=3 feet). She draws the curve starting at the origin and ending at the point (44, 443/2). How many inches of chalk does Annie have remaining after she draws this curve?

Probably 1.112

Sylvia the potter has just completed a large order of ornate clay bowls but has just realized that she is one short. She only has 1 cubic foot of clay left. Does she have enough to make the last bowl? (The bowl can be modeled by region bounded by y=x2 and y=√x when rotated around the x-axis.)

Yes. 0.5 left.

A strange and unusual particle moves with velocity equal to v(t)=t10*ln(t) m/s. You observe this particle carefully in a lab and find that over the time interval [1, e1/11] the particle moves _______ meters.

1/121

An old man with a bad back wants to hang a plane mirror on his wall and wants to know what angle it will make with the floor so that he can lie down on his back under it and watch TV from across the room. The problem is, his house was designed by Frank Gehry and the wall is modeled by the equation z=½x2+¼y2. He wants to attach the mirror to the wall tangent to the point (2,4,6). What angle will this tangent mirror (plane) make with the floor? (You can assume that the floor is represented by the equation z=0.)

70.5 degrees.

A factory producing computers has two factors that contribute to production: labor (L) and capital (K). The number of computers they can produce in an hour is given by the function P=100L0.4K0.6. Unfortunately, the factory's budget is $3000 per hour and one unit of labor costs $75 per hour and one unit of capital costs $100 per hour. To produce the most computers in one hour, how many units of labor should the factory employ and how many units of capital should they use? Write your answer as the ordered pair L,K where L is the number of units of labor and K is the number of units of capital.

16.18. Most likely wrong..

You go out for dinner at Papa Raboloid's Italian Bistro and order his signature 16π drink. He pours 16π cubic centimeters of reddish liquid into your paraboloidic cup. (Your cup can be modeled by the equation z=2x2+2y2.) Your cup is 10 cm tall. How many centimeters will be left unfilled at the top of the cup?

2

Can someone check and tell me that I got all wrong?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hi;

I am 13 now and am in my second year of university

Congratulations! That is very good.

First question about the deer:

I think this is what your equation looks like. I hope I have translated it to latex correctly.

Using L' Hospitals rule you will end up with, d(t) --> 75 as t --> ∞.

second question, Annie and the chalk:

If you mean

as the curve then you are correct.

Fourth question, unusual particle:

Same as you got.

If you are not correct with your answer then at least we will both be wrong.

**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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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

Well, I have to hand it in by tomorrow so I need to check the other questions as well. Anyways, thanks a lot for the help. You, along with Ganesh seem to be the only active members.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Ummm... Guys... Shouldn't the answer to the first one be the maximum value of d(t)?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,393

When in doubt try looking at the graph.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hi anonimnystefy;

That is the maximum value.

**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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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

Hi Stefy (mind if I call you that);

I believe that the max value of d(t) is the answer.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

bobbym wrote:

Hi anonimnystefy;

That is the maximum value.

It is, but you would hqve tocprove that too.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

You prove it by taking the limit of that function.

**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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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Not true. That just gets you the limit, but doesn't guarantee that that value is the maximum of the function.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

That is very good and true. There is a way to determine where the maximum value is and prove that it occurs at infinity.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Yes, and that is what is needed to be done in order for the proof to be complete.

Here lies the reader who will never open this book. He is forever dead.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

Well... Using L'Hopitals Rule, we simply take the derivative of the numerator and the denominator separately to get

. Then simply calculate the limit by cancelling the terms to receive the final answer 75.Offline

**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

By the way, I apologize for my improper LaTeX syntax.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hi Shivamcoder3013;

What anonimnystefy is saying is just because you go to infinity that does not mean you have found a maximum. There is a way to prove have the largest value at 75.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

Oh, okay. By the way, you can call me Shivam or Cless. Anyways, I didn't go to Yale today so I will submit my results later.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

bobbym calls everybody by their forum username.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Are you sure about that?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,525

Yes. You said so yourself.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Yes, I always try to use the username. It is the name the person chose to sign in here so it is obviously there preferred moniker.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

I submitted my results today at Yale and recieved an almost-perfect mark. Apparently, my answer to the particle question, despite the fact that Bobbym agreed with me, is incorrect.

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,532

The solution provided by my professor is 1/131, with no direct clarification of the answer.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,676

Hi;

Looks like a typo.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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