The Volume of a tin of radius r cm is given by V=pi*(40r-r^2-r^3). Find the positive value of r for which dV/dr=0 and find the value of V for this r.

Firstly differentiate the function V with respect to x (dV/dx)=pi*(40-2r-3r^2). Set dV/dr =0 and solve to find r. Divide both sides by pi and divide both side by -1 so that the r^2 term is positive (I personally find it easier to solve when the highest power coefficient is positive) (3r^2+2r-40=0). Factorise this expression ((3r-10)(r+4)=0). From this you can deduce the two values of x for which dV/dx=0 (r=10/3 and r =-4). Lastly substitute the positive value of r (10/3) into the expression for V (V=pi*(40-2(10/3)-3(10/3)^2)) = (2300*pi)/27.