Question

The height and the slant height of a cone are 21cm and 28 cm respectively. Find the volume if the cone.​

Answer 1

Answer:

The volume of the cone is 7546 cm³.

Step-by-step-explanation:

We have given that,

• Height ( h ) of cone = 21 cm
• Slant height ( l ) of cone = 28 cm

We have to find the volume of the cone.

We know that,

$\pink{\sf\:l^2\:=\:h^2\:+\:r^2}\sf\:\:-\:-\:[\:Formula\:]$

$\implies\sf\:r^2\:=\:l^2\:-\:h^2$

$\implies\sf\:r^2\:=\:(\:28\:)^2\:-\:(\:21\:)^2$

$\implies\sf\:r^2\:=\:(\:28\:+\:21\:)\:(\:28\:-\:21\:)\:\:-\:-\:[\:\because\:a^2\:-\:b^2\:=\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:]$

$\implies\sf\:r^2\:=\:49\:\times\:7$

$\implies\sf\:r\:=\:7\:\times\:\sqrt{7}\:\:\:-\:-\:[\:Taking\:square\:roots\:]$

$\implies\boxed{\red{\sf\:r\:=\:7\:\sqrt{7}\:cm}}\sf\:\:-\:-\:(\:1\:)$

Now, we know that,

$\pink{\sf\:Volume\:of\:cone\:=\:\dfrac{\pi\:r^2\:h}{3}}\sf\:\:\:-\:-\:[\:Formula\:]$

$\displaystyle\implies\:\sf\:V_{cone}\:=\:\dfrac{\dfrac{22}{7}\:\times\:7\:\sqrt{7}\:\times\:7\:\sqrt{7}\:\times\:21}{3}\:\:-\:-\:[\:From\:(\:1\:)\:]$

$\implies\sf\:V_{cone}\:=\:\dfrac{22}{\cancel7}\:\times\:7\:\sqrt{7}\:\times\:7\:\sqrt{7}\:\times\:\cancel{21}\:\times\:\dfrac{1}{3}$

$\implies\sf\:V_{cone}\:=\:\dfrac{22\:\times\:7\:\sqrt{7}\:\times\:7\:\sqrt{7}\:\times\:\cancel{3}}{\cancel{3}}$

$\implies\sf\:V_{cone}\:=\:22\:\times\:7\:\sqrt{7}\:\times\:7\:\sqrt{7}$

$\implies\sf\:V_{cone}\:=\:22\:\times\:49\:\times\:7$

$\implies\sf\:V_{cone}\:=\:22\:\times\:343$

$\implies\boxed{\red{\sf\:Volume\:of\:cone\:=\:7546\:cm^3}}$

$\rule{200}{1}$

Additional Information:

1. Cone:

Any three dimensional figure having two surfaces with base circular in shape is called as cone.

2. Examples of conical objects:

Conical tent, ice - cream cone, sharpened end of pencil, etc are some examples of conical objects.

3. Important formulae related to cone:

A cone having height $\sf\:h$, slant height $\sf\:l$ and radius $\sf\:r$ has following formulae:

$\displaystyle\sf\:1.\:Area\:of\:base\:=\:\pi\:r^{2}\\\\\sf\:2.\:l^{2}\:=\:r^{2}\:+\:h^{2}\:\:\:\:or\:\:\:l\:=\:\sqrt{r^{2}\:+\:h^{2}}\\\\\sf\:3.\:Curved\:surface\:area\:=\:\pi\:r\:l\\\\\sf\:4.\:Total\:surface\:area\:=\:\pi\:r\:(\:r\:+\:l\:)\\\\\sf\:5.\:Volume\:=\:\dfrac{1}{3}\:\pi\:r^{2}\:h$

Answer 2

⭐GIVEN:

• Slant height of cone (l) = 28cm

• Height of a cone (h) = 21cm

⭐TO FIND:

• Volume of a cone=?

⭐SOLUTION:

Now, let's find the radius of cone.

: . l²=h²+r².......... (Formula)

: . (28) ²=(21) ²+r²

: . 784=441+r²

: . r²=784-441

: . r²=343

: . r=√343

: . r=√7×49

: . r=7√7 cm

Therefore,The radius of a cone is 7√7 cm.

• VOLUME OF A CONE=1/3×πr²h

=1/3×22/7×(7√7)²×21

=22/21×343×21

=22/3×49×21

=22×49×7

=7546 cm³

Therefore,The volume of a cone is 7546cm³.......