[tex]v=1.5l*l*w[/tex]
[tex]w=6-l-1.5l[/tex]
[tex]v=1.5l*l*(6-l-1.5l)[/tex]
which simplifies to
[tex]v=9l^2-3.75l^3[/tex]
and so we take the derivative of that function in terms of l
[tex]\frac{d}{dx}v=18l-11.25l^2[/tex]
then we set that to 0
[tex]-11.25l^2+18l=0[/tex]
using the quadratic formula
[tex]\frac{-b+or- \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\frac{-18+or- \sqrt{18^2-4*11.25*0}}{2*-11.25}[/tex]
simplifying..
[tex]\frac{-18+or- \sqrt{324}}{-22.5}[/tex]
[tex]\frac{-18+\sqrt{324}}{-22.5}=0[/tex]
[tex]\frac{-18-\sqrt{324}}{-22.5}=1.6[/tex]
at this point i noticed an error, i used l instead of h... but thats ok
pluggin 0 and 1.6 for h, lets check answers
1.6: h=1.6 l=2.4 w=2
0: h=0 l=0 w=6
so the answer is h=1.6 l=2.4 w=2, which gives the greatest volume of 7.68
