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#1 2016-12-04 03:39:06

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Minimum surface area

Hi. My snail speed math progression has carried me to a bunch of word exercises in the precalculus book that i'm trying to go through for most of this year, and i am pitifully terrible at word problems:

The box for the new Sasquatch-themed cereal, ‘Crypt-Os’, is to have a volume of 140 cubic inches. For aesthetic reasons, the height of the box needs to be 1.62 times the width of the base of the box. Find the dimensions of the box which will minimize the surface area of the box. What is the minimum surface area? Round your answers to two decimal places.

So my attempt at solving this is
1) find height function (i don't know another way of finding this without making width and length equal):
        h = 1.62x
        x^2 * 1.62x = 140
        h = 140/1.62x^2
2) find surface area function and plug in the height function:
        2x^2 + 4xh
        2x^2 + 4x(140/1.62x^2)
        2x^2 + 560x/1.62x^2
3) I feel like i should find the slant of this function, then find the vertex of this slant, which will give me x, then i should plug that number in surface area function, and get the minimal surface area. Of course none of this is working out for me, so could anyone lend me a helping hand here?

Last edited by sisyphus (2016-12-04 04:03:34)

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#2 2016-12-04 04:15:56

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimum surface area

Hi;

You need to minimize the surface area 2(hw)+2(lh)+2(wl) subject to the constraints that lwh=140 and h=1.62w. This leads to a definite answer.

Oh and by the way, people see the legendary Sasquatch here often. It goes by the name Skunk Ape. No one really knows what the creature eats. Speculation says, it will grab a gator and snap it into two pieces as effortlessly as we could a twig. Others believe it kills bears for its grub. The darker side people think he will eat a human when it can. Nobody, but nobody, thinks it eats cereal.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2016-12-04 04:46:06

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Minimum surface area


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#4 2016-12-04 05:11:20

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Re: Minimum surface area

bobbym wrote:

Hi;

You need to minimize the surface area 2(hw)+2(lh)+2(wl) subject to the constraints that lwh=140 and h=1.62w. This leads to a definite answer.

Oh and by the way, people see the legendary Sasquatch here often. It goes by the name Skunk Ape. No one really knows what the creature eats. Speculation says, it will grab a gator and snap it into two pieces as effortlessly as we could a twig. Others believe it kills bears for its grub. The darker side people think he will eat a human when it can. Nobody, but nobody, thinks it eats cereal.

Thanks. I tried it likes this now:

lwh = 140
h = 1.62w
1.62lw^2 = 140
l = 140/1.62w^2

2(hw)+2(lh)+2(wl)
2(1.62w^2)+2(1.62lw)+2(wl)=
2(1.62w^2)+2*1.62w(140/1.62w^2)+2w(140/1.62w^2)=
1.62w^2+3.24w(140/1.62w^2)+2w(140/1.62w^2)=
(1.62w^3+452.8395)/w~

So now i'm getting w = -6.54~. If i would take it as a positive i'm still no where near the 4.12~ listed in the book answers. Is my method still false, or have i made some arithmetic mistake?

Last edited by sisyphus (2016-12-04 05:14:06)

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#5 2016-12-04 06:37:27

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimum surface area

Hi;

A) S = 2*h*w + 2 l*h + 2 w*l

Using the fact that h = (81 / 50) w we substitute and get

B) S = (131 l w)/25 + (81 w^2)/25

Solving for l in l*w*h = 140 we get

l = 7000/(81 w^2)

Substituting that into B) we get:

S = 36680/(81 w) + (81 w^2)/25

Now all you have to do is set the derivative of that equal to 0 and solve for w. What do you get now? The problem with this approach is that you will have to solve a cubic. Easy for a computer, hard for a human. And, 4.12 is correct for w rounded to 2 decimal places.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#6 2016-12-04 07:31:17

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Re: Minimum surface area

bobbym wrote:

Hi;

A) S = 2*h*w + 2 l*h + 2 w*l

Using the fact that h = (81 / 50) w we substitute and get

B) S = (131 l w)/25 + (81 w^2)/25

Solving for l in l*w*h = 140 we get

l = 7000/(81 w^2)

Substituting that into B) we get:

S = 36680/(81 w) + (81 w^2)/25

Now all you have to do is set the derivative of that equal to 0 and solve for w. What do you get now? The problem with this approach is that you will have to solve a cubic. Easy for a computer, hard for a human. And, 4.12 is correct for w rounded to 2 decimal places.

1) But i solved it the same way, i just did not use fractions (i might have made one error with the number in front of w^3)? The part i don't understand is (6561w^3 + 917000)/2025w. Am i supposed to factor the numerator into a quadratic, because that's the only thing i sort of know how to do with cubics? You mentioned derivatives, but the book does not even have a word like that in it. Is this something i am supposed to know by this point?

By the way, i am using this book http://wp.vcu.edu/precalculus/files/2013/08/Precalculus-3rd-ed.pdf (i'm currently at page 354)

2) Also, how can i improve the way i present my solutions (or rather attempts at them), so that helpful people like you could easier understand what the hell i was trying to do. I always try to show what i tried to do, but it seems to me like i'm doing such a big mess that nobody can understand what i wrote down. The obvious answer for this topic is that i could have used fractions, but other than that, where i am confusing?

Last edited by sisyphus (2016-12-04 07:37:35)

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#7 2016-12-04 08:53:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimum surface area

What did you get for an answer? Did you get 164.91 for the minimum surface area?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#8 2016-12-04 11:08:33

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Re: Minimum surface area

bobbym wrote:

What did you get for an answer? Did you get 164.91 for the minimum surface area?

Sorry for my language barrier, i used the word "solved" to mean that i got to (6561w^3 + 917000)/2025w. I did not actually get an answer since i don't understand how to proceed further. At first i though that i would just get rid of the denominator and solve for w, but you mentioned something about derivatives, which i know nothing about. I tried to factor the numerator, but 6561 is not an integer multiple of 917000, so trying out thousands of compatible fractions would be a bit of a difficult task.

Last edited by sisyphus (2016-12-04 11:22:06)

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#9 2016-12-04 13:31:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimum surface area

Hi;

You can get right answer with your S = (6561 w^3 + 917000)/(2025 w) but my S = 36680/(81 w) + (81 w^2)/25 is a bit easier too look at.

Either way you now need to minimize S. That will involve the derivative. If you do not know what that is then we can minimize using iteration but that might be confusing also. There might be another way using the AM-GM inequality but that too would be confusing.

Looking at the book you are using that fellow uses a graphic calculator on his earlier problems. You are trying to do the problems without one of those? For that you will need a bit more math than you currently know. The graphic calculator way does not require you to know so much math. The pencil and paper way requires that you do.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#10 2016-12-04 18:44:12

sisyphus
Member
Registered: 2016-05-15
Posts: 26

Re: Minimum surface area

Ah, i now remember the author mentioning that about those sort of problems now. Then i will not dwindle on this and continue on with the book.

Thank you bobbym.

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#11 2016-12-04 19:19:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Minimum surface area

Okay, post when you get another one or when you are ready to see how to do it.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#12 2016-12-04 19:24:42

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Minimum surface area

sisyphus wrote:

Hi. My snail speed math progression has carried me to a bunch of word exercises in the precalculus book that i'm trying to go through for most of this year, and i am pitifully terrible at word problems:

The box for the new Sasquatch-themed cereal, ‘Crypt-Os’, is to have a volume of 140 cubic inches. For aesthetic reasons, the height of the box needs to be 1.62 times the width of the base of the box. Find the dimensions of the box which will minimize the surface area of the box. What is the minimum surface area? Round your answers to two decimal places.

So my attempt at solving this is
1) find height function (i don't know another way of finding this without making width and length equal):
        h = 1.62x
        x^2 * 1.62x = 140
        h = 140/1.62x^2
2) find surface area function and plug in the height function:
        2x^2 + 4xh
        2x^2 + 4x(140/1.62x^2)
        2x^2 + 560x/1.62x^2
3) I feel like i should find the slant of this function, then find the vertex of this slant, which will give me x, then i should plug that number in surface area function, and get the minimal surface area. Of course none of this is working out for me, so could anyone lend me a helping hand here?


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Offline

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