The total surface area of a solid cylinder is 1155 cm$$^2$$. Its curved surface area is two-fifth of its total surface area. What is the height(in cm) of the cylinder?(Take $$\pi = \frac{22}{7}$$)

The total surface area of a solid cylinder is 1155 cm$$^2$$.

the total surface area of a solid cylinder = 1155

$$2\times\ \pi\ \times\ r\times\ \left(r+h\right) = 1155$$    Eq.(i)

Here r = radius and h = height.

Its curved surface area is two-fifth of its total surface area.

$$2\times\ \pi\ \times\ r\times h=\frac{2}{5}\times\ 2\times\ \pi\ \times\ r\times\ \left(r+h\right)$$

$$h=\frac{2}{5}\times\left(r+h\right)$$

5h = 2r+2h

5h-2h = 2r

3h = 2r

h : r = 2 : 3

Let's assume h = 2y and r = 3y.

Put the value of 'h' and 'r' in Eq.(i).

$$2\times\ \pi\ \times\ 3y\times\ \left(3y+2y\right)=1155$$

$$2\times\ \frac{22}{7}\times\ 3y\times5y=1155$$

$$2\times\ \frac{2}{7}\times\ 3y\times5y=105$$

$$\frac{4}{7}\times\ 15y^2=105$$

$$\frac{4}{7}\times y^2=7$$

$$y^2=7\times\ \frac{7}{4}$$

$$y^2=\left(\frac{7}{2}\right)^2$$

$$y=\frac{7}{2}$$

Height of the cylinder = 2y = $$2\times\frac{7}{2}$$ = 7 cm