Question

A cone of height 24 cm has a curved surface area of 550 cm^2. Find it's volume.

Answer 1

Csa of cone =pie *r*l
550=22/7*r*24
550*7/22*24=r
r=3850/528
r=7.2
Volume of cone =1/3*pie *r^2 *h
=1/3*22/7*7.2*7.2*24
=27371.5/21
=1303.4 ans

Hope it is helpful and mark me as brainliest if you satisfied from my answer

Answer 2

Given:

• Height of cone = 24 cm
• Curved Surface Area of cone = 550 cm²

Need to find:

• Volume of cone

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━

☯ Let r cm be the radius of base and l cm the slant height.

$\setlength{\unitlength}{1cm}\begin{picture}(6, 4)\linethickness{0.26mm} \qbezier(5.8,2.0)(5.8,2.3728)(4.9799,2.6364)\qbezier(4.9799,2.6364)(4.1598,2.9)(3.0,2.9)\qbezier(3.0,2.9)(1.8402,2.9)(1.0201,2.6364)\qbezier(1.0201,2.6364)(0.2,2.3728)(0.2,2.0)\qbezier(0.2,2.0)(0.2,1.6272)(1.0201,1.3636)\qbezier(1.0201,1.3636)(1.8402,1.1)(3.0,1.1)\qbezier(3.0,1.1)(4.1598,1.1)(4.9799,1.3636)\qbezier(4.9799,1.3636)(5.8,1.6272)(5.8,2.0)\put(0.2,2){\line(1,0){2.8}}\put(3.2,4){\sf{24 cm}}\put(5,4){\sf{l cm}}\put(3,2){\line(0,2){4.5}}\put(1.4,1.6){\sf{r cm}}\qbezier(.185,2.05)(.7,3)(3,6.5)\qbezier(5.8,2.05)(5.3,3)(3,6.5)\put(3,2.02){\circle*{0.15}}\put(2.7,2){\dashbox{0.01}(.3,.3)}\end{picture}$

⠀⠀

Therefore,

$\qquad\sf l^2 = r^2 + 24^2\qquad\qquad\bigg\lgroup\bf Using\; l^2 = r^2 + h^2\bigg\rgroup$

$:\implies\sf l^2 = r^2 + 576$

$:\implies\sf \sqrt{l^2} = \sqrt{r^2 + 576}$

$:\implies\sf l = \sqrt{r^2 + 576}$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━

Now,

We know that,

$\sf\star\; CSA\;of\;cone = 550\;cm^2$

$\qquad\quad:\implies\sf \pi rl = 550$

$:\implies\sf \dfrac{22}{7} \times r \times \sqrt{r^2 + 576} = 550$

$:\implies\sf r \sqrt{r^2 + 576} = 550 \times \dfrac{7}{22}$

$:\implies\sf r \sqrt{r^2 + 576} = 25 \times 7$

$:\implies\sf r^2(r^2 + 576) = (25 \times 7)^2$

$:\implies\sf r^4 + 576r^2 - (25^2 \times 7^2) = 0$

$:\implies\sf r^4 + 576r^2 - (625 \times 49) = 0$

$:\implies\sf r^4 + 625r^2 - 49r^2 - 625 \times 49 = 0$

$:\implies\sf r^2(r^2 + 625) - 49(r^2 + 625) = 0$

$:\implies\sf (r^2 + 625)(r^2 + 625) = 0$

$:\implies\sf r^2 - 49 = 0\qquad\qquad\bigg\lgroup\bf \because\; r^2 + 625 \neq 0\bigg\rgroup$

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

$:\implies\sf \sqrt{r^2} = \sqrt{49}$

$:\implies{\boxed{\frak{\pink{r = 7\;cm}}}}\;\bigstar$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━

Now, Finding Volume of cone,

$\star\;{\boxed{\sf{\purple{Volume_{\;(cone)} = \dfrac{1}{3} \pi r^2h}}}}$

$:\implies\sf \dfrac{1}{ \cancel{3}} \times \dfrac{22}{ \cancel{7}} \times \cancel{7} \times 7 \times \cancel{24}$

$:\implies{\boxed{\frak{\pink{1232\;cm^2}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Hence,\;Volume\;of\;cone\;is\; \bf{1232\;cm^3}.}}}$