**Question:**

A solid cylinder has a total surface area of $231 \mathrm{~cm}^{2}$. Its curved surface area is $\frac{2}{3}$ of the total surface area. Find the volume of the cylinder.

**Solution:**

We know that the total surface area of the cylinder is 231 cm2 and the curved surface area is 2/3 of the total surface area.

So, the curved surface area is:

$2 / 3 \times\left(231 \mathrm{~cm}^{2}\right)=154 \mathrm{~cm}^{2}$

Then, the radius of the cylinder can be calculated in the following manner:

Curved surface area = 2π*rh*

154 cm2 = 2π*rh** *... (1)

Here,* r* cm is the radius of the cylinder and *h* cm is the length of the cylinder.

2πr2 = (231-154) cm2 = 77 cm2

77 cm2 = 2π*r*2

From here, the radius *(r*) can be calculated in the following manner:

$\boldsymbol{r}=\sqrt{\frac{77}{2 \times \frac{22}{7}}}$

r = 3.5 cm

Substituting this result into equation (1):

1

54 cm2 = 2π(3.5 cm)*h*

$h=154 \mathrm{~cm}^{2} /\left(2 \times \frac{22}{7} \times(3.5 \mathrm{~cm})\right)$

*h* = 7 cm

$\therefore V=\pi r^{2} h=\frac{22}{7} \times(3.5 \mathrm{~cm})^{2} \times(7 \mathrm{~cm})=269.5 \mathrm{~cm}^{3}$

Hence, the volume of the cylinder is 269.5 cm3.

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