Question

The volume of a right circular cylinder is 3 times the volume of a right circular cone . theadius of the cone and the cylinder are 3 cm and 6cm respectively . if the height of the cylinder is 1 cm , them whatis the slant height of the cone ? A.
$\sqrt{13cm}$
B. 4 CM
C. 5 CM
D.
$\sqrt{15cm}$
​

Answer 1

Answer:

Option (c) 5 cm.

Step-by-step explanation:

Given :-

• The volume of the right circular cylinder is 3 times the volume of the right circular cone.
• Radius of the cone and the cylinder are 3 cm and 6 cm respectively.
• The height of the cylinder is 1 cm.

To find :-

• Slant height of the cone.

Solution :-

Formula used :

★ ${\boxed{\sf{Volume\:of\: cylinder=\pi\:r^2h}}}$

★ ${\boxed{\sf{Volume\:of\:cone=\dfrac{1}{3}\pi\:r^2h}}}$

In case of cylinder ,

• Radius= 6 cm
• Height = 1 cm

Then,

Volume of the cylinder,

= πr²h

= (π × 6² × 1 ) cm³

= 36π cm³

In case of cone,

• Radius = 3 cm

Let the height of the cone be h cm and the slant height of the cone be l cm.

Volume of the cone,

$\sf =( \dfrac{1}{3} \times \pi \times {3}^{2} \times h) \: {cm}^{3} \\ \\ = \sf \: 3\pi \: h \: {cm}^{3}$

According to the question ,

Volume of cylinder = 3 ×Volume of cone

→ 36π = 3×3πh

→ 12 = 3h

→ h = 4

Height of the cone is 4 cm.

Now find the slant height of the cone .

l² = h² + r²

→ l² = 4²+3²

→ l² = 16+9

→ l² = 25

→ l = 5

Therefore, the slant height of the cone is 5 cm.

Answer 2

Solution :

$\underline{\bf{Given\::}}$

• The volume of a right circular cylinder is 3 times the volume of a right circular cone.
• The radius of the cone = 3 cm
• The radius of the cylinder = 6 cm
• Height of the cylinder = 1 cm

$\underline{\bf{Explanation\::}}$

As we know that formula of the volume of right circular cylinder & cone;

$\boxed{\bf{Volume\:_{(cylinder)} = \pi r^{2} h \:\:(cubic\:unit)}}\\\boxed{\bf{Volume\:_{(cone)} =1/3 \pi r^{2} h \:\:(cubic\:unit)}}$

A/q

$\mapsto\sf{Volume\:_{(cylinder)} = 3\times Volume\:_{(cone)} }\\\\ \mapsto \sf{\pi r^{2} h = 1/3\pi r^{2} h } \\\\\mapsto\sf{ \cancel{22/7 } \times (6)^{2} \times 1 = 3\times 1/3 \times \cancel{22/7} \times (3)^{2} \times h}\\\\\mapsto\sf{36 = \cancel{3} \times 1/\cancel{3} \times 9 \times h}\\\\\mapsto\sf{36 = 9h}\\\\\mapsto\sf{h=\cancel{36/9}}\\\\\mapsto\bf{h=4\:cm}$

Now;

As we know that formula of the slant height of cone;

$\boxed{\bf{Slant\:height\:(l) = \sqrt{r^{2} + h^{2}} }}$

$\mapsto\sf{Slant\:height\:_{(cone)} = \sqrt{(3)^{2} + (4)^{2} } }\\\\\mapsto\sf{Slant\:height\:_{(cone)} = \sqrt{9 + 16 }}\\\\\mapsto\sf{Slant\:height\:_{(cone)} = \sqrt{25}}\\\\\mapsto\bf{Slant\:height\:_{(cone)} = 5\:cm}$

Thus;

The slant height of the cone will be 5 cm .