Determine the r-values?

Question

Let V and S denote the volume and total surface area, respectively, of a right circular cylinder of radius r and height 1 unit. For which r-values will the ratio V/S be less than 1/3?

Answer 1

Best Answer

(πr^2h)/(2πrh+2πr^2)<1/3

(πr^2)/(2πr+2πr^2)<1/3

(πr^2)/2πr(1+r)<1/3

r/2(1+r)<1/3

r<1/3(2+2r)

r<2/3+2r/3

r-2r/3<2/3

r(1-2/3)<2/3

r<(2/3)/(1-2/3)

r<6/3

r<2

Answer 2

So V = pi*(r^2)*h = pi*r^2

and S = 2*pi*r*h + 2*pi*r^2 = 2*pi*r + 2*pi*r^2 = 2*pi*r*(1+r)

so V/S = pi*r^2 / 2*pi*r*(1+r) = r/(2(1+r))

when is this less than 1/3?

r/(2(1+r)) < 1/3

3r/(2(1+r)) <1

3r<2(1+r)

3r<2+2r

r<2

and obviously r has to be positive so

0<r<2