Question

the total surface area of right circular cone with slant height 25 cm is 704 cm sq,find the volume of the cone

Answer 1

Step-by-step explanation:

Total surface area of right circular cone = 704cm²

Slant height (l) = 25cm

Radius of the cone (r) = ?

Total surface area of cone = πrl + πr²

⇢ πrl + πr² = 704

⇢ πr ( l + r ) = 704

⇢ r ( l + r) = $\dfrac{704}{\pi}$

⇢ r ( 25 + r) = $704 \times \dfrac{7}{22}$

⇢ 25r + r² = 224

⇢ r² + 25r - 224 = 0

⇢ r² + 32r - 7r - 224 = 0

⇢ r( r + 32) - 7( r + 32) = 0

⇢ ( r - 7) ( r + 32) = 0

⇢ r - 7 = 0 (or) r + 32 = 0

⇢ r = 7 (or) r = -32

Since, radius cannot be negative

Radius of the cone (r) = 7cm

Height (h) = $\sqrt{l^{2} - {r}^{2}}$

$\rm \dashrightarrow h = \sqrt{ 25^{2} - 7^{2}}$

$\rm \dashrightarrow h = \sqrt{ 625 - 49 }$

$\rm \dashrightarrow h = \sqrt{576}$

$\rm \dashrightarrow h = 24cm$

Now, Volume of the cone = $\rm \dfrac{1}{3} \pi r^{2} h$

= $\rm \dfrac{1}{3} \times \dfrac{22}{7} \times 7^{2} \times 24$

= $\rm \dfrac{22}{7} \ times 7 \times 7 \times 8$

= 22 × 7 × 8

= 1232

$\underline{\boxed{\rm \therefore Volume \: of \: the \: cone = 1232cm^{3}}}$