Changing dimensions in a rectangular box. Suppose that the edge lengths x, y and z of a closed rectangular box are changing at the following rates :
`(dx)/(dt)=1m//"sec", (dy)/(dt)=-2m//"sec", (dy)/(dt)=1//"sec"`,
Find the rates at which the box's (a) volume, (b) surface area and (c) diagonal length `s=sqrt(x^(2)+y^(2)+z^(2))` are changing at the instant when x=4, y=3 and z=2 .

Explanation

`V= xyz`
`(dV)/(dt)=y=(dx)/(dt)+xz(dy)/(dt)+xy(dz)/(dt)`
`=6xx1+8xx (-2)+12(1)`
`=6-16+12=2`
`S=2(lb+bh+lh)`
`(dS)/(dt)=2[(l+b)(dh)/(dt)+(bh)(dl)/(dt)+(hl)(db)/(dt)]`
`=2[7(1)+5(1)+6(-2)]`
=0
`l=sqrt(x^(2)+y^(2)+z^(2))`
`"So" , " " (dl)/(dt)=(2x(dx)/(dt)+2y(dy)/(dt)+2z(dz)/(dt))/(2 sqrt(x^(2)+y^(2)+z^(2)))`
`=(4(1)+3(-2)+2(1))/(sqrt(4^(2)+3^(2)+2^(2)))=0`