Question

A solid consists of a circular cylinder surmounted by a right circular cone. The height of the cone is h. If the total volume of the solid is 3 times the volume of the cone, then the height of the cylinder is
(a)2h
(b)$\frac{3h}{2}$
(c)$\frac{h}{2}$
(d)$\frac{2h}{3}$

Answer 1

Answer:

The height of the cylinder is ⅔ h.

Among the given options option (d) ⅔ h is the correct answer.

Step-by-step explanation:

Given :  

The height of the cone is h  

Let the height of the cylinder be x .

Volume of the total solid = 3 × volume of the cone

(Volume of circular cylinder  + volume of right circular cone) = 3 × volume of the cone

πr²h + 1/3πr²h = 3 × (1/3πr²h)

πr²x + 1/3πr²h = πr²h

πr²x = πr²h - 1/3πr²h

πr²x = (3πr²h - 1πr²h)/3

πr²x = 2/3πr²h

x = ⅔ h

Height of the cylinder = ⅔ h

Hence, the height of the cylinder is ⅔ h.

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Answer 2

Answer:

The height of the cylindrical = $\frac{2h}{3}$

Among the options given, Option d) $\frac{2h}{3}$ is the correct option.

Step-by-step explanation:

Let radius of the cone and cylinder be r units.

Height of the cone = h units.

Let Height of the cylinder be H units.

According to the Question,

Total volume of the solid = 3 × Volume of the cone

⇒ $\frac{1}{3}$πr²h + πr²H = 3 × $\frac{1}{3}$πr²h

⇒ πr²H = πr²h - $\frac{1}{3}$πr²h

⇒ πr²H = πr²h ( 1 - $\frac{1}{3}$ )

Cancelling πr² on both sides

⇒ H = h × ( $\frac{3\:-\:1}{3}$ )

⇒ H = h × ( $\frac{2h}{3}$ )

⇒ H = $\frac{2h}{3}$

$<b><u>•°• The height of the cylinder = 2h / 3</u></b>$