Question

The Volume, Vcm3, of a tin of radius r cm is given by the formula V=π (40r-r2-r3). Find the positive value or r for which dV/dr=0, and find the value of V which corresponds to this value of r.

Answer 1

`V=\pi (40r-r^2-r^3) \\V=40\pi r-\pi r^2-\pi r^3 \\\\ (dV)/(dr)=40\pi -2\pi r-3\pi r^2 \\\\ (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`

r is positive, so r =10/3 cm

So, if r =10/3:


`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.)`