Question 1013915
Height h
Diameter d
v for volume


{{{system(v=(1/4)h*pi*d^2,h+d=24)}}}


{{{v=(1/4)pi*(24-d)*d^2}}}
{{{v=(1/4)pi(24d^2-d^3)}}}
{{{v=6pi*d^2-(pi/4)d^3}}}


A calculus practice, derivative:
{{{dv/dd=12pi*d-(3pi/4)d^2}}}


{{{12pi*d-(3pi/4)d^2=0}}}
{{{d(12pi-(3pi/4)d)=0}}}
{{{d=0,NotReasonable}}}
-
{{{12pi=(3pi/4)d}}}
{{{d=12*4*pi/(3pi)}}}
{{{d=4*4}}}
{{{highlight(d=16)}}}
and the corresponding height, {{{highlight(h=8)}}}.


Those should give the maximum volume using that restriction for h and d.