rectangular box ( maxima)

[ZyzzBrah]

An open box is to be made by cutting out squares from the corners of a rectangular piece of cardboard and then turning up the sides. if the piece of cardboard is 12 by 24 in , what are the dimensions of the box of largest volume made this way?

Soln:
The sides are
s1 = 12 - 2y
s2 = 24 - 2y
s3 = y

V = (12 - 2y)(24 - 2y)(y)
\(\displaystyle V = 4y^3 -72y^2 + 288y \)

\(\displaystyle \frac{dV}{dt} = 12y^2 -144y + 288 = 0\)

is y = 2.536

is this correct?

 

[soroban]

Hello, ZyzzBrah!

Correct! . . . Excellent work!

 

[srmichael]

An open box is to be made by cutting out squares from the corners of a rectangular piece of cardboard and then turning up the sides. if the piece of cardboard is 12 by 24 in , what are the dimensions of the box of largest volume made this way?

Soln:
The sides are
s1 = 12 - 2y
s2 = 24 - 2y
s3 = y

V = (12 - 2y)(24 - 2y)(y)
\(\displaystyle V = 4y^3 -72y^2 + 288y \)

\(\displaystyle \frac{dV}{dt} = 12y^2 -144y + 288 = 0\)

is y = 2.536

is this correct?

Yes, that is correct! You must have just done related rates recently because you took the derivative with respect to t. That is obviously not the case in optimization problems.