Determine the dimensions of a rectangular box, open at the top

[naspek]

Homework Statement

Determine the dimensions of a rectangular box, open at the top, having volume
4 m3, and requiring the least amount of material for its construction. Use the
second partials test. (Hint: Take advantage of the symmetry of the problem.)

Homework Equations

The Attempt at a Solution

Volume, V= 4m^3
let x = length
y = width
z = height

4m^3 = xyz
x = 4/yz

because it is an open at the top rectangular box,
the Surface, S = 2xz + 2yz + xy

substitute x=4/yz inside the surface equation..
S = 8/y + 2yz + 4/z

to find the critical points, take the partial with respect to y and z.. the equal it to zero..

S'(y)= 2z - 8/y^2
S'(z)= 2y - 4/z^2

solve the equations, i get,
when y = 0, z = 0..
y = 8, z = 1/16

the problem is, what should i do next?

 

[Mark44]

When y = 0, 8/y^2 is undefined! When I set S_y and S_z to 0, I get different values than you did.

Also, you're supposed to use the second partials test.

 

[naspek]

ic... ok..

so..

S'(y) = 2z - 8/y^2
S'(yy) = 16/y^3
S'(yz) = 2

S'(z) = 2y - 4/z^2
S'(zz) = 8/z^3
S'(zy) = 2

what should i do next?

 

[Mark44]

You still haven't found the values of y and z for which S_y = 0 and S_z = 0.

Also, you haven't used the second partials test; all you have done is calculate all the second partials.

 

[naspek]

ok.. for.. S'(y)=0 -----> 2z - 8/y^2=0

...for.. S'(z)=0 -----> 2y - 4/z^2 =

ok.. by letting z=4/y^2, i get y = 0, +8^1/2, -8^1/2

am i got it correct now?

 

[naspek]

is it possible if i use lagrange multiplier method?

 

[naspek]

done with the equation for x, y and z..
got some miscalculation just now..
i got y=2, z= 1, x = 2..
why i should do the second partial test?

 

[Mark44]

done with the equation for x, y and z..
got some miscalculation just now..
i got y=2, z= 1, x = 2..
why i should do the second partial test?

Those are the values I got.

You should use the second partial test for two reasons:

1. The problem asks you to use it.
2. So you can tell whether these values give you a maximum value or a minimum value of S.

 

[naspek]

thanks.. u are really helpful =)