Question

Froma right circular cylinder of radius 7 cm, height 24cm of conical cavity of same base radius and of same height hollowed out. find the volume and whole surface of remaining solid ​

Answer 1

$\huge\red{\boxed{\sf AnSwer}}$

Given :-

Height of cylinder/cone (h)= 24cm

Radius of cylinder/cone (r)= 7cm

To find:-

We have to find out the Volume and surface area of remaining solid

Solution:-

Volume of remaining solid = Volume of cylinder- Volume of cone.

$\begin{gathered}:\implies\sf\ V.\ of\ remaining\ solid = \pi r^2 h-\dfrac{1}{3}\pi r^2\ h \\ \\ \\ :\implies\sf\ V.\ of\ remaining\ solid = \dfrac{1}{3}\pi r^2 h\big(3-1\big)\\ \\ \\ :\implies\sf\ V.\ of\ remaining\ solid = \dfrac{1}{\cancel3}\times \dfrac{22}{\cancel{7}}\times \cancel{7}\times 7\times \cancel{24}\times \big(2\big)\\ \\ \\ :\implies\sf\ V.\ of\ remaining\ solid =22\times 7\times 8\times 2\\ \\ \\ :\implies\underline{\boxed{\pink{\sf\ Volume\ of\ remaining\ solid = 2624cm^3}}}\end{gathered}$

● Surface area of remaining solid = CSA of cylinder+ CSA of cone + Area of top (circular part)

$\sf\bigstar\ \ Surface\ Area= 2\pi r h+ \pi r \ell + \pi r^2$

$\begin{gathered}\bullet\sf \ell= \sqrt{r^2+h^2}\\ \\ \longmapsto\sf \ell= \sqrt{(7)^2+(24)^2}\\ \\ \longmapsto\sf \ell= \sqrt{49+576}\\ \\ \longmapsto\sf \ell= \sqrt{625}\\ \\ \underline{\boxed{\sf\ \ell= 25cm}}\end{gathered}$

Now surface Area :-

$\begin{gathered}:\implies\sf\ Surface\ Area= \pi r\big\lgroup 2h+\ell+r\big\rgroup\\ \\ \\ :\implies\sf\ Surface\ Area= \dfrac{22}{\cancel{7}}\times \cancel{7}\big\lgroup (2\times 24)+ 25+7\big\rgroup\\ \\ \\ :\implies\sf\ Surface\ Area= 22\times \big\lgroup 48+32\big\rgroup\\ \\ \\ :\implies\sf\ Surface\ Area= 22\times 80\\ \\ \\ :\implies\underline{\boxed{\purple{\sf\ Surface\ area\ of\ remaining\ solid= 1760cm^2}}}\end{gathered}$

Answer 2

Answer:

\huge\red{\boxed{\sf AnSwer}}

AnSwer

Given :-

Height of cylinder/cone (h)= 24cm

Radius of cylinder/cone (r)= 7cm

To find:-

We have to find out the Volume and surface area of remaining solid

Solution:-

Volume of remaining solid = Volume of cylinder- Volume of cone.

\begin{gathered}\begin{gathered}:\implies\sf\ V.\ of\ remaining\ solid = \pi r^2 h-\dfrac{1}{3}\pi r^2\ h \\ \\ \\ :\implies\sf\ V.\ of\ remaining\ solid = \dfrac{1}{3}\pi r^2 h\big(3-1\big)\\ \\ \\ :\implies\sf\ V.\ of\ remaining\ solid = \dfrac{1}{\cancel3}\times \dfrac{22}{\cancel{7}}\times \cancel{7}\times 7\times \cancel{24}\times \big(2\big)\\ \\ \\ :\implies\sf\ V.\ of\ remaining\ solid =22\times 7\times 8\times 2\\ \\ \\ :\implies\underline{\boxed{\pink{\sf\ Volume\ of\ remaining\ solid = 2624cm^3}}}\end{gathered}\end{gathered}

:⟹ V. of remaining solid=πr

2

h−

3

1

πr

2

h

:⟹ V. of remaining solid=

3

1

πr

2

h(3−1)

:⟹ V. of remaining solid=

3

1

×

7

22

×

7

×7×

24

×(2)

:⟹ V. of remaining solid=22×7×8×2

:⟹

Volume of remaining solid=2624cm

3

● Surface area of remaining solid = CSA of cylinder+ CSA of cone + Area of top (circular part)

\sf\bigstar\ \ Surface\ Area= 2\pi r h+ \pi r \ell + \pi r^2★ Surface Area=2πrh+πrℓ+πr

2

\begin{gathered}\begin{gathered}\bullet\sf \ell= \sqrt{r^2+h^2}\\ \\ \longmapsto\sf \ell= \sqrt{(7)^2+(24)^2}\\ \\ \longmapsto\sf \ell= \sqrt{49+576}\\ \\ \longmapsto\sf \ell= \sqrt{625}\\ \\ \underline{\boxed{\sf\ \ell= 25cm}}\end{gathered}\end{gathered}

∙ℓ=

r

2

+h

2

⟼ℓ=

(7)

2

+(24)

2

⟼ℓ=

49+576

⟼ℓ=

625

ℓ=25cm

Now surface Area :-

\begin{gathered}\begin{gathered}:\implies\sf\ Surface\ Area= \pi r\big\lgroup 2h+\ell+r\big\rgroup\\ \\ \\ :\implies\sf\ Surface\ Area= \dfrac{22}{\cancel{7}}\times \cancel{7}\big\lgroup (2\times 24)+ 25+7\big\rgroup\\ \\ \\ :\implies\sf\ Surface\ Area= 22\times \big\lgroup 48+32\big\rgroup\\ \\ \\ :\implies\sf\ Surface\ Area= 22\times 80\\ \\ \\ :\implies\underline{\boxed{\purple{\sf\ Surface\ area\ of\ remaining\ solid= 1760cm^2}}}\end{gathered}\end{gathered}

:⟹ Surface Area=πr

⎩

⎪

⎧

2h+ℓ+r

⎭

⎪

⎫

:⟹ Surface Area=

7

22

×

7

⎩

⎪

⎧

(2×24)+25+7

⎭

⎪

⎫

:⟹ Surface Area=22×

⎩

⎪

⎧

48+32

⎭

⎪

⎫

:⟹ Surface Area=22×80

:⟹

Surface area of remaining solid=1760cm

2