Math, asked by preetishroff8, 2 months ago

The area of the base of a conical solid is 2464 cm2

and its volume is 17248 cm3

. Find the

curved surface area of the solid.

Answers

Answered by Anonymous
144

Appropriate Question :-

  • The area of the base of a conical solid is 2464 cm² and its volume is 17248 cm³. Find the curved surface area of the solid.

Given :-

  • The area of the base of a conical solid is 2464 cm² and its volume is 17248 cm³.

To Find :-

  • What is the curved surface area of the solid.

Formula Used :-

\clubsuit Volume Of Cone Formula :

\mapsto \sf\boxed{\bold{\pink{Volume_{(Cone)} =\: \dfrac{1}{3}{\pi}r^2h}}}

\clubsuit Area of Circle Formula :

\mapsto \sf\boxed{\bold{\pink{Area_{(Circle)} =\: {\pi}r^2}}}\\

\clubsuit Curved Surface Area Of Cone Formula :

\mapsto \sf\boxed{\bold{\pink{C.S.A_{(Cone)} =\: {\pi}rl}}}\\

where,

  • π = pie or 22/7
  • r = Radius
  • h = Height
  • l = Slant Height

Solution :-

First, we have to find the radius :

Given :

  • Area = 2464 cm²

According to the question by using the formula we get,

\implies \sf \dfrac{22}{7} \times r^2 =\: 2464

\implies \sf r^2 =\: \dfrac{2464 \times 7}{22}

\implies \sf r^2 =\: \dfrac{\cancel{17248}}{\cancel{22}}

\implies \sf r^2 =\: 784

\implies \sf r =\: \sqrt{784}

\implies \sf\bold{\green{r =\: 28\: cm}}

Now, we have to find the height :

Given :

  • Volume = 17248 cm³
  • Radius = 28 cm

According to the question by using the formula we get,

\implies \sf \dfrac{1}{3} \times \dfrac{22}{7} \times (28)^2 \times h =\: 17248

\implies \sf \dfrac{22}{21} \times 784 \times h =\: 17248

\implies \sf \dfrac{17248}{21} \times h =\: 17248

\implies \sf h =\: \dfrac{\cancel{17248} \times 21}{\cancel{17248}}

\implies \sf \bold{\green{h =\: 21\: cm}}

Now, we have to find the value of slant height :

As we know that,

\clubsuit Pythagoras Theorem Formula :

\mapsto \sf\boxed{\bold{\pink{l =\: \sqrt{r^2 + h^2}}}}\\

where,

  • l = Slant Height
  • r = Radius
  • h = Height

Given :

  • Radius = 28 cm
  • Height = 21 cm

According to the question by using the formula we get,

\implies \sf l =\: \sqrt{(28)^2 + (21)^2}\\

\implies \sf l =\: \sqrt{784 + 441}

\implies \sf l =\: \sqrt{1225}

\implies \sf\bold{\green{l =\: 35\: cm}}

Now, we have to find the curved surface area of the cone :

Given :

  • Radius = 28 cm
  • Slant height = 35 cm

According to the question by using the formula we get,

\longrightarrow \sf C.S.A_{(Cone)} =\: \dfrac{22}{7} \times 28 \times 35\\

\longrightarrow \sf C.S.A_{(Cone)} =\: \dfrac{22}{7} \times 980

\longrightarrow \sf C.S.A_{(Cone)} =\: \dfrac{21560}{7}

\longrightarrow \sf\bold{\red{C.S.A_{(Cone)} =\: 3080\: cm^2}}

\therefore The curved surface area or CSA of cone is 3080 cm².

Answered by SparklingBoy
122

\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese  \:  \:  \: GIVEN \:  \:  \:  \maltese }}}}

 \large \mathfrak{ \text{F}or  \:  \:  \text{A}  \:  \: Conical \:  \:  \text{ S}olid}

  \bf Base \: area = 2464 \:  {cm}^{2}   \\  \\  \bf Volume = 17248 {cm}^{3}

 \color{magenta}\large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese  \:  \:  \:To \:  \: Calculate \:  \:  \:  \maltese }}}}

 \bf CSA \:  of \:  the \:  Conical\:Solid

 \color{green} \large\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese  \:  \:  \: SOLUTION \:  \:  \:  \maltese }}}}

  \huge\mathfrak{Let }  \\ \mathfrak{ \: \: Radius \:  \:  of \:  \:  Base \:  \:  of  \:  \:  \text{S}olid} = \bf r \\  \\  \mathfrak{ \text{H}eight \:  \: of \:  \:   \text{S}olid} =  \bf \: h

 \huge \mathcal{As} \\  \\  \bf \: Base \:  \:  Area = \pi {r}^{2}  = 2464 \\  \\  \implies \sf \frac{22}{7}  \times  {r}^{2}  = 2464 \\  \\  \implies \sf  {r}^{2}  = 2464 \times  \frac{7}{22}   \\  \\  =  112 \times 7  \\  \\  \implies \sf \:  {r}^{2}  = 784  \\  \\ \implies \bf {r}^{2}  =  \sqrt{784}  \\ \\ \implies\color{red} \boxed{ \boxed{ \bf r = 28 \: cm}}

  \huge\mathcal{NOW}

 \bf Volume = \frac{1}{3}\times\pi {r}^{2} h  = 17248 \\  \\  \implies \sf \frac{1}{3}\times\frac{22}{7}  \times784 \times h = 17248 \\  \\  \implies \sf h =  \frac{17248\times3}{2464}  \\  \\  \implies \bf  \color{purple}\boxed{ \boxed{ \bf h = 21 \: cm}}

Now,

Let Slant height = l

\sf l= \sqrt{h^2+r^2} \\ \\ \sf l=\sqrt{441+784}\\ \\ \bf \color{orange} \boxed{ \boxed{ \bf l = 35 \: cm}}

 \huge \mathfrak{ \text{F}inall \text{y}}

 \bf CSA = \pi r l \\  \\  =   \frac{22}{7}  \times 28 \times 35 \\  \\ \color{green} \boxed{ \boxed{ \bf CSA = 3080 {cm}^{2} }}

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