Question

A solid cylinder and a solid cone have equal bases and equal heights. If the radius and height be in the ratio 4 : 3, the ratio of the total surface area of the cylinder to that of the cone is __________.

Answer 1

Let r be the radius of the solid cylinder and solid cone & h be the height of the solid cylinder and solid cone.

It is given that, r : h = 4 : 3

Suppose r = 4x units and h = 3x units, where x is constant

Let l be the slant height of the cone.

$\therefore l=\sqrt{r^2+h^2}=\sqrt{{\left(4x\right)}^2+{\left(3x\right)}^2}=\sqrt{25x^2}=5x$ units

Now,

Total surface area of the cylinder = 2$\mathrm{\pi }$r(r + h) = 2$\mathrm{\pi }$ × 4x × (4x + 3x) = 56$\mathrm{\pi }$x² sq. units

Total surface area of the cone = $\mathrm{\pi }$r(r + l) = $\mathrm{\pi }$ × 4x × (4x + 5x) = 36$\mathrm{\pi }$x² sq. units

$\therefore \frac{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cylinder}}{\mathrm{Total}\mathrm{surface}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{cone}}=\frac{56\mathrm{\pi }{x}^2}{36\mathrm{\pi }{x}^2}=\frac{14}{9}$

Thus, the ratio of total surface area of the cylinder to the total surface area of the cone is 14 : 9.

A solid cylinder and a solid cone have equal bases and equal heights. If the radius and height be in the ratio 4 : 3, the ratio of the total surface area of the cylinder to that of the cone is ___14 : 9___.