[tex]V=\pi (40r-r^2-r^3)
\\V=40\pi r-\pi r^2-\pi r^3
\\\\ \frac{dV}{dr}=40\pi -2\pi r-3\pi r^2
\\\\ \frac{dV}{dr}=0
\\40\pi -2\pi r-3\pi r^2=0
\\40-2r-3r^2 = 0
\\3r^2+2r-40 = 0
\\3r^2+12r-10r-40 = 0
\\3r(r+4)-10(r+4)=0
\\(3r-10)(r+4)
\\r=-4,10/3[/tex]
r is positive, so r =10/3 cm
So, if r =10/3:
[tex]V=\pi (40r-r^2-r^3)
\\V = \pi(40(10/3)-(10/3)^2-(10/3)^3)
\\V=\pi(400/3-100/9-1000/27)
\\V =\pi(85.185...)
\\V = 85.185...\pi
\\V = 267.617...
\\V = 268 cm^3 (3 s.f.)[/tex]
