Asked by Suman Koirala
on 26 Mar 2013

Problem 1: The volume V and paper surface area of a conical paper cup are given by:

V=1/3*pi*r^2*h

A =pi*r*sqrt(r^2+h^2)

For V = 10 in 3 , compute the value of the radius, r that minimizes the area A. What is the corresponding value of the height, h? What is the minimum amount that r can vary from its optimal value before the area increases by 10%.

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Answer by Youssef Khmou
on 26 Mar 2013

Edited by Youssef Khmou
on 26 Mar 2013

Accepted answer

hi Suman Koirla, try this :

The Volume is given by : V=(1/3)*r²*h, and the surface A=pi*r*sqrt(r²+h²)

for V=10 m^3, we search for r that minimizes the Surface , :

Min(A) , SUBject to V=10

we have : h=3*V/pi*r² then : A=pi*r*sqrt(r²+90/pi²*r^4) .

Min(A) means the dA/dr=0=......=4*pi*r^3-180 /(2*sqrt(pi*r^4+90/r²))=0

Fast way to find R :

syms r A=(pi^2*r^2+90/r^2)^1/2 ezplot(A) S=subs(A,-6:0.1:6); % AXIS based on the first graph min(S)

1)so the minimum value for S=29.83 meter is R=1.89 ( FROM THE GRapH )

2) The corresponding value for h=3*10/(pi*1.89)=5.0525 meter .

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Walter Roberson
on 26 Mar 2013

No, not h=3*V/pi*r² -- h=3*V/(pi*r²)

The actual minimum value for r is 1.890102955

Youssef Khmou
on 26 Mar 2013

YES true i made mistake its S=29 m², corresponding r~1.9 meter .

Suman Koirala
on 26 Mar 2013

How to do the third part where it says "What is the minimum amount that r can vary from its optimal value before the area increases by 10%." I had no idea on that one. Thanks for any inputs.

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Answer by Walter Roberson
on 26 Mar 2013

Are you required to use a minimizer? The question can be solved analytically with a tiny amount of algebra together with some small calculus.

Suman Koirala
on 26 Mar 2013

Not required to use minimizer. Intro Matlab course.

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Answer by Youssef Khmou
on 27 Mar 2013

Edited by Youssef Khmou
on 27 Mar 2013

3)What is the minimum amount that r can vary from its optimal value before the area increases by 10% ( with fixed h ) :

Given S=29.83 m² and h=5.05 m, we have the new surface S2 :

__________ S2=S+0.1*S=32.81 m²=pi*r*\/ r²+h² .

S2²=pi².r^4 + pi²r²h² , make it as equation of 4th order :

r^4 + r² . h² -S2²/pi² = 0 ==> r^4 + 25.50 *r² - 109.7 = 0

We use the command "root" :

the Polynomial is a*r^4 + b*r^3 + c*r^2 + b*r + d = 0

a=1; b=0; c=25.50; d=-109.7

R_amount = roots([1 0 25.50 0 -109.7]) R_amount =

0.0000 + 5.4084i 0.0000 - 5.4084i 1.9366 -1.9366

The reasonable answer is the third one, R=1.9366 the amount change is

DELTA_R=1.9366-1.89=0.04 meter .

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## 6 Comments

## Suman Koirala (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/68627-finding-minimum-value-of-radius#comment_138948

I have done this so far:

## Image Analyst (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/68627-finding-minimum-value-of-radius#comment_138955

What does "10 in 3" mean?

## Youssef Khmou (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/68627-finding-minimum-value-of-radius#comment_138959

i think, it means for V=10 in "equation 3" , maybe

## Walter Roberson (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/68627-finding-minimum-value-of-radius#comment_138960

You have asked fminbnd() to invoke your function 'Untitled3', which then will invoke fminbnd() which will then invoke Untitled3, which will then invoke fminbnd()...

## Walter Roberson (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/68627-finding-minimum-value-of-radius#comment_138962

I wonder if "10 in 3" is intended to mean "10 cubic inches" ?

## Suman Koirala (view profile)

Direct link to this comment:https://www.mathworks.com/matlabcentral/answers/68627-finding-minimum-value-of-radius#comment_138964

Hey guys, sorry for that..it was 10 cubic inches.

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