\(v=1.5l*l*w\)
\(w=6-l-1.5l\)
\(v=1.5l*l*(6-l-1.5l)\)
which simplifies to
\(v=9l^2-3.75l^3\)
and so we take the derivative of that function in terms of l
\(\frac{d}{dx}v=18l-11.25l^2\)
then we set that to 0
\(-11.25l^2+18l=0\)
using the quadratic formula
\(\frac{-b+or- \sqrt{b^2-4ac}}{2a}\)
\(\frac{-18+or- \sqrt{18^2-4*11.25*0}}{2*-11.25}\)
simplifying..
\(\frac{-18+or- \sqrt{324}}{-22.5}\)
\(\frac{-18+\sqrt{324}}{-22.5}=0\)
\(\frac{-18-\sqrt{324}}{-22.5}=1.6\)
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
