Ex.13.2 Q4 Surface Areas and Volumes Solution - NCERT Maths Class 10

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Question

A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are \(15 \,\rm{cm}\) by \(10 \,\rm{cm} \) by \(3.5 \,\rm{cm}\). The radius of each of the depressions is \(0.5\,\rm{cm}\) and the depth is \(1.4\,\rm{cm}\).

Find the volume of wood in the entire stand (see Fig. 13.16).

 

Text Solution

 

What is known?

A wooden pen stand is in the shape of a cuboid with four conical depressions.

The dimensions of the cuboid are \(15\rm\,cm\; \times 10\,cm \;\times\; 3.5\,cm\)  

Radius of conical depressions is \(\,0.5\,\rm{cm}\)

Depth of conical depression is \(1.4\,\rm{cm}\).

What is unknown?

Volume of wood in the entire pen stand.

Reasoning:

From the given figure it’s clear that the conical depressions do not contain wood. Since the dimensions of all \(4 \)conical depressions are the same, they will have identical volumes too.

 Volume of wood in the entire pen stand \(=\) volume of the wooden cuboid \( - {\rm{ }}4 \times \)volume of the conical depression

 We will find the volume of the solid by using formulae;

Volume of the cuboid\( = lbh\)

where \(l, b\) and \(h\) are the length, breadth and height of the cuboid respectively.

Volume of the cone\(\begin{align} = \frac{1}{3}\pi {r^2}{h_1}\end{align}\)

where \(r\) and \(h1\) are the radius and height of the cone respectively

Steps:

Depth of each conical depression,\({h_1} = 1.4cm\)

Radius of each conical depression, \(r = 0.5cm\)

Dimensions of the cuboid are \(15cm \times 10cm \times 3.5cm\)

Volume of wood in the entire pen stand \(=\) Volume of the wooden cuboid \(- 4 \;\times \) Volume of each conical depression

\[\begin{align}&= lbh - 4 \times \frac{1}{3}\pi {r^2}{h_1}\\ &= \left( {15cm \times 10cm \times 3.5cm} \right) - \left( {4 \times \frac{1}{3} \times \frac{{22}}{7} \times 0.5cm \times 0.5cm \times 1.4cm} \right)\\&= 525c{m^3} - 1.47c{m^3}\\&= 523.53c{m^3}\end{align}\]

  
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