Question

A hemispherical dome is on a circular room. Its total internal volume is 48510 m^3 and inner diameter is equal to maximum height. Find the height of the room.

Answer 1

Given:

A hemispherical dome is on a circular room.

Its total internal volume is 48510 m³ and inner diameter is equal to maximum height.

To find:

The height of the room

Solution:

Let's assume,

"H" → represents the maximum height of the room

"r" → represents the inner radius.

Since the inner diameter is equal to maximum height, so

H = d

∴ Inner radius, r = $\frac{d}{2} = \frac{H}{2}$

∴ Height of the cylindrical part, h = H - r = $H - \frac{H}{2} = \frac{H}{2}$

Now, we have

[Internal volume of the room] = [Internal volume of the hemispherical dome] + [Internal volume of the cylindrical part]

⇒ [Internal volume of the room] = [$\frac{2}{3} \pi r^3$] + [$\pi r^2 h$]

on substituting the values, we get

⇒ $48510 = [\frac{2}{3}\times \frac{22}{7}\times (\frac{H}{2} )^3] + [\frac{22}{7} \times (\frac{H}{2} )^2 \times \frac{H}{2}]$

⇒ $48510 = [\frac{22}{7} \times (\frac{H}{2} )^3 ][\frac{2}{3}+1]$

⇒ $48510 = [\frac{22}{7} \times (\frac{H}{2} )^3 ][\frac{2}{3}+1]$

⇒ $48510 = \frac{22}{7} \times (\frac{H}{2} )^3 \times \frac{5}{3}$

⇒ $48510 = \frac{22}{7} \times \frac{H^3}{8} \times \frac{5}{3}$

⇒ $H^3 = \frac{48510 \times 168}{110}$

⇒ $H = \sqrt[3]{\frac{48510 \times 168}{110}}$

⇒ $H = \sqrt[3]{74088}$

⇒ $\bold{H = 42\:m}$

Thus, the height of the room is → 42 m.

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Answer 2

Step-by-step explanation:

The height is 42 cm.

This is very important questions for Class 10th