A cylinder and a cone have equal radii of their bases and equal height

1.	A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8 : 5, Show that the radius of each is to the height of each as 3 : 4.
Answer» For cylinder, we have Base radius = r Height = h ∴ Curved surface area S₁ = 2πrh For cone, we have Slant height l = \(\sqrt{r^2+h^2}\) S₂ = πrl = πr\(\sqrt{r^2+h^2}\) We have, \(\frac{S_1}{S_2}\) = \(\frac{8}{5}\) = \(\frac{2\pi rh}{\pi r\sqrt{r^2+h^2}}\) = \(\frac{2h}{\sqrt{r^2+h^2}}\) = \(\frac{8}{5}\) (squaring both side) = \(\frac{4h^2}{{r^2+h^2}}\) = \(\frac{64}{25}\) = 100h²= 64r²+ 64h² = 36h² = 64r² (Square root both side) = 6h = 8r = \(\frac{r}{h}\) = \(\frac{3}{4}\)

A cylinder and a cone have equal radii of their bases and equal heights. If their curved surface areas are in the ratio 8 : 5, Show that the radius of each is to the height of each as 3 : 4.

Answer»

For cylinder, we have

Base radius = r

Height = h

∴ Curved surface area S₁ = 2πrh

For cone, we have

Slant height l = \(\sqrt{r^2+h^2}\)

S₂ = πrl = πr\(\sqrt{r^2+h^2}\)

We have,

\(\frac{S_1}{S_2}\) = \(\frac{8}{5}\) = \(\frac{2\pi rh}{\pi r\sqrt{r^2+h^2}}\)

= \(\frac{2h}{\sqrt{r^2+h^2}}\) = \(\frac{8}{5}\)

(squaring both side)

= \(\frac{4h^2}{{r^2+h^2}}\) = \(\frac{64}{25}\)

= 100h²= 64r²+ 64h²

= 36h² = 64r²

(Square root both side)

= 6h = 8r

= \(\frac{r}{h}\) = \(\frac{3}{4}\)