The given figure shows a frustum. The radius of the upper base, the radius of the lower base, and the height of the frustum are in the ratio of 2:5:4 respectively. The curved surface area of the frustum is 990 cm². What is the height of the frustum?

It is given that the radius of the upper base, the radius of the lower base, and the height of the frustum are in the ratio of 2:5:4.

Let the radius (r₁) of the upper base of the frustum be 2x.

Thus, radius (r₂) of the lower base of the frustum = 5x

Thus, height (h) of the frustum = 4x

Slant height (l) of the given frustum = $\sqrt{{\left(r_2-r_1\right)}^2+h^2}=\sqrt{{\left(5x-2x\right)}^2+{\left(4x\right)}^2}=5x$

Curved surface area (A) of the frustum = $990 c{m}^2=\mathrm{\pi }({\mathrm{r}}_1+{\mathrm{r}}_2)l$

$\Rightarrow 990=\frac{22}{7}\times \left(2x+5x\right)5x$

$\Rightarrow 990=\frac{22}{7}\times 7x\times 5x$

$\Rightarrow x^2=9$

$\Rightarrow x=3$

Thus, height of the given frustum = 4x = 4×3 cm = 12 cm