Question

A pen stand is made of wood in the shape of cuboid with three conical depressions to hold the pens. The dimensions of the cuboid are \[15\text{ }cm\text{ }by\text{ }10\text{ }cm\text{ }by~3.5\text{ }cm\]. The radius of each of the depression is \[0.5\text{ }cm\] and the depth is \[1.4\text{ }cm\]. Find the volume of wood in the entire stand.

Answer 1

Hint: The idea is to subtract the volume of the depressions from the total volume of the cuboid to get the volume of wood in the stand.

Complete step-by-step answer:
Here the length of cuboid = 15cm, breadth = 10cm and the height = 3.5cm as given in the question.

The total volume of the depressions can be calculated by using the formula for Volume of cone which is \[\dfrac{1}{3}*\pi *{{r}^{2}}*h\]
Here the radius of each cone = 0.5cm and height = 1.4cm.

To get the volume of wood we need to subtract the volume of depressions from the total volume of the cuboid.

Volume of wood = Volume of cuboid – Volume of depressions

Volume of wood = \[l*b*h=\dfrac{1}{3}*\pi *{{r}^{2}}*h\]

Hence substituting these values in the expression for volume of cuboid and volume of depressions we find the volume of wood material :


  Volume Of Cuboid=$l*b*h
 =15*10*3.5=525c{{m}^{3}}$
 Volume Of 3 Conical Depressions$=3*\dfrac{1}{3}*\pi *{{r}^{2}}*h
 =\pi *{{0.5}^{2}}*1.4=1.1c{{m}^{3}} $
 Volume Of Wood=$525-1.1
 =529.3c{{m}^{3}}$

Note: Make sure to apply the correct values in the formulae appropriately for each dimension. Here the point to note is we are taking out the volume of conical depressions as the depressions won’t have any wood in them. Therefore the true volume of wood remaining in the cuboid can be found by subtraction of the values of the volume of total conical depressions from the total volume of the wooden cuboid.