Right Circular Cylinder

Right Circular Cylinder

A cylinder whose base is a circle is called circular cylinder. If the axis of the cylinder is perpendicular to its base then the cylinder is called a right circular cylinder.

If $$r$$ is the radius of the circular base, $$h$$ is the height, $$S$$ is the lateral surface area, $$V$$ is the volume and $$P$$ is the perimeter of the base, then

$$Volume\,of\,the\,cylinder\, = \,area\,of\,base\, \times \,height\,of\,cylinder$$
i.e        (i)                     $$V = \pi {r^2}h$$     ($$r$$ being radius)
or         (ii)                    $$V = \frac{\pi }{4}{d^2}h$$          ($$d$$ being diameter)

 

Example:

Find the cost of digging a well 3m in diameter and 24m deep at the rate of Rs.10 per cu.m.

 

Solution:

Given that

            Diameter of the well       $$ = 3m$$
$$\therefore $$            Radius of the well,      $$r = 1.5m$$
Depth of the well,       $$h = 24m$$
$$\therefore $$            Volume of the earth excavated           $$ = \pi {r^2}h = \frac{{22}}{7} \times \left( {1.52} \right) \times 24$$
$$ = 3.1416 \times 2.25 \times 24\,\,cu.\,m$$
$$ = 169.646\,\,cu.\,m$$
Now, cost per cu. m    $$ = {\text{Rs}}{\text{.}}\,10$$
$$\therefore $$            $${\text{Total}}\,{\text{Cost}}\, = \,10 \times 169.646 = 1696.46\,{\text{rupees}}$$

 

Example:

The volume of a cylindrical ring is 800 cu.cm. The radius of a cross section is 2cm. Find the length of the ring.

 

Solution:

Given that

            Radius of cross section,          $$r = 2\,{\text{cm}}$$
Volume,                                  $$v = 800\,{\text{cu}}{\text{.cm}}$$

            Let the length of the ring be   $$h\,\,\,cm$$
Now,   $${\text{Volume}}\, = \,{\text{area of cross – section }} \times {\text{ height}}$$
$$800\,\,\,\,\,\,\, = \pi {r^2} \times h$$
$$ = \frac{{22}}{7} \times 4 \times h$$

            $$\therefore $$            $$h = \frac{{800 \times 7}}{{22 \times 7}} = \frac{{700}}{{11}} = 63.6\,{\text{cm}}$$