Question #9ab18

May 7, 2016

$137984 c {m}^{3}$

Explanation:

Let the radius of the right circular cylinder be r cm
and height be h cm

The area of its Curved Surface =$2 \pi r h$ cm^2

The area of its Plane Crcular Surface =$2 \pi {r}^{2}$ cm^2

So The area of its Total Surface =$\left(2 \pi {r}^{2} + 2 \pi r h\right)$ cm^2

By the 1st condition of the problem:
The ratio of total surface area to the curved surface area of a right circular cylinder is 3 : 2
Hence
$\frac{2 \pi {r}^{2} + 2 \pi r h}{2 \pi r h} = \frac{3}{2} \implies \left(\frac{\cancel{2 \pi r} \left(r + h\right)}{\cancel{2 \pi r} h}\right) = \frac{3}{2}$
$\implies \frac{r + h}{h} = \frac{3}{2}$
$\implies 3 h = 2 r + 2 h$
$\implies h = 2 r$

Now by the 2nd condition of the problem:

Total surface area = 14784 cm^2
So $\left(2 \pi {r}^{2} + 2 \pi r h\right) = 14784$
$\implies \left(2 \pi {r}^{2} + 2 \pi r \cdot 2 r\right) = 14784$ [ putting h=2r ]
$\implies \left(2 \pi {r}^{2} + 4 \pi {r}^{2}\right) = 14784$
$\implies \left(6 \pi {r}^{2}\right) = 14784$
$\implies \left(6 \cdot \frac{22}{7} {r}^{2}\right) = 14784$

$\implies {r}^{2} = {\cancel{14784}}^{112} \cdot \frac{7}{\cancel{22}} \cdot \frac{1}{\cancel{6}} = 16 \cdot 7 \cdot 7$
$r = \sqrt{{4}^{2} \cdot {7}^{2}} = 28 c m$

Hence Radius of the Cylinder = r=28 cm

So its height $h = 2 r = 2 \times 28 c m = 56 c m$

and Volume of the Cylinder =$\pi {r}^{2} h = \frac{22}{7} \cdot {28}^{2} \cdot 56 c {m}^{3}$
$= \frac{22}{\cancel{7}} \cdot {\cancel{28}}^{4} \cdot 28 \cdot 56 c {m}^{3} = 137984 c {m}^{3}$