Volume of Cylinder

A cylindrical container has a base radius of 7 cm and a depth of 10 cm. Find the volume if $\mathrm{\pi }=\frac{22}{7}$

Solution:

We first note the radius and the height of the cylinder, $r=7cm,h=10cm$.

The volume of the circular cylinder is calculated as,

Find the volume of the figure below, taking $\pi =\frac{22}{7}.$

Solution:

Recalling Cavalier's principle,

$V_{obliquecylinder}=V_{circularcylinder}=\pi r^{2}h$

We deduce from the figure that$r=9cm,h=28cm$.

Thus, the volume of the oblique cylinder given in the above figure can be calculated as,

$V_{obliquecylinder}=\frac{22}{7}\times 9^2\times 28=22\times 81\times 4=7128c{m}^3$.

Find the volume of a semicircular cylinder with a height of 6 cm and a diameter of 5 cm. Take $\pi =\frac{22}{7}.$

Solution:

The volume of a semicircular cylinder is given by,

$V_{semicircularcylinder}=\frac{\pi r^2\times h}{2}$

We write down the height and the diameter from the given,$h=6cm,d=5cm$.

We deduce the radius from the diameter, $r=\frac{diameter}{2}=\frac{5}{2}cm.$

Hence, the volume of the semicircular cylinder is given by,

$V_{semicircularcylinder}=\frac{\pi r^2\times h}{2}=\frac{\pi \times {\left(\frac{5}{2}\right)}^2\times 6}{2}=\frac{\frac{22}{7}\times \frac{25}{4}\times 6}{2}=\frac{\frac{3300}{28}}{2}=58.93c{m}^3$.

Determine the volume of the casket below. Take $\pi =\frac{22}{7}.$

Solution:

We first note that the top of the casket is a semicircular cylinder while the base is a rectangular prism.

Let us find the volume of the semicircular cylindrical top.

$V_{semicircularcylinder}=\frac{\pi r^2\times h}{2}$

We note that the diameter of the semicircular cylinder is $d=14cm.$Thus, $r=\frac{diameter}{2}=\frac{d}{2}=\frac{14}{2}=7cm.$

Hence,

$V_{semicircularcylinder}=\frac{\pi r^2\times h}{2}=\frac{\frac{22}{7}\times 7^2\times 30}{2}=\frac{22\times 7\times 30}{2}=2310c{m}^3$.

The volume of the rectangular prism,

$V_{rec\tan gularprism}=length\times breadth\times heightoftheprism$

From the figure, we deduce that length = 30 cm, breadth = 14 cm and height = 15 cm.

Hence,

$V_{rec\tan gularprism}=30\times 14\times 15=6300c{m}^3.$

The volume of the casket is calculated as the sum of the volume of the semicircular cylinder and the volume of the rectangular prism.

$V_{casket}=V_{semicircularcylinder}+V_{rec\tan gularprism}=2310+6300=8610c{m}^3$.

How many tissue rolls does Brenda need to block 40 425 cubic centimeters opening in her room if the height of the roll is 50 cm? Take $\pi =\frac{22}{7}.$

Solution:

To determine how many rolls of tissues Brenda has to use, we need to find the volume of the tissue, $V_{tissue}$.

The volume of the tissue can be calculated by subtracting the volume of the tissue's hollow space, from the volume of the whole cylinder.

Thus,

$V_{tissue}=V_{wholecylinder}-V_{hollowspace}$

We calculate first the volume of the whole cylinder,

$V_{wholecylinder}=\mathrm{\pi }\times {\mathrm{r}}^2\times \mathrm{h}=\mathrm{\pi }\times {\left(\frac{28}{2}\right)}^2\times 50=\frac{22}{7}\times {14}^2\times 50=30800{\mathrm{cm}}^3$

Next, in order to calculate the volume of the hollow space, we first need to calculate its corresponding radius. But the diameter of the hollow space can be found by subtracting the diameter of the whole cylinder from the diameter of the non-empty cylinder, thus

$diamete{r}_{hollowcylinder}=28-7=21cm$

Now, the volume of the hollow space is,

$V_{hollowspace}=\pi \times r^2\times h=\frac{22}{7}\times {\left(\frac{21}{2}\right)}^2\times 50=17325{\mathrm{cm}}^3.$

Thus the volume of the tissue is,

$V_{tissue}=V_{wholecylinder}-V_{hollowspace}=30800-17325=13475c{m}^3.$

Since the volume of the space Brenda is to fill is 40 425 cm³, then she would need,

$(40425÷13475)tissues=3tissues$.