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 cm by 10 cm by 3.5 cm. The radius of each of the depression is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand.

Answer 1

Formula used :

$\boxed{\large\mathbf{Volume \: of \: Cuboid = lbh}}$

$\boxed{\boxed{\large\mathbf{Volume \: of \: Cone = \frac{1}{3}π{r}^{2}h}}}$

Given,

The Dimensions of the cuboidal pen stand is 15cm × 10cm × 3.5cm.
The radius of the conical depression is 0.5cm.
Height of the conical depression is 1.4cm.

$\underline{\underline{\huge\mathfrak{Solution}}}$

Let us first calculate the Volume of Cuboidal stand:

$\underline{\mathbf{Volume \: of \: Cuboidal \: stand}}$

$= 15 \times 10 \times 3.5 \\ \bf = 525 {cm}^{2}$

$\underline{\mathbf{Volume \: of \: 3 \: Conical \: depression}}$

$= 3( \frac{1}{3} \pi {r}^{2} h) \\ \\ = 3( \frac{1}{3} \times \frac{22}{7} \times {(0.5)}^{2} \times 1.4) \\ \\ \bf can \: also \: be \: written \: as \\ \\ = \frac{3}{3} \times \frac{22}{7} \times \frac{5}{10} \times \frac{5}{10} \times \frac{14}{10} \\ \\ = \frac{11}{10} \\ \\ \bf = 1.1 {cm}^{2}$

The volume of the wood in the entire stand = Volume of cuboidal stand - (Volume of three conical depression)

$= 525 - 1.1 \\ \bf = 523.9 {cm}^{2}$

If any doubt, please ask! ;)

Answer 2

$\huge\textbf\green{Answer :}$

Given :-

$\textbf{Dimensions of cuboid = 15 cm, 10 cm and 3.5 cm}$

$\textbf{Radius of each dimensions are = 0.5 cm}$

$\textbf{Height of each depression = 1.5 cm}$

$\textbf{We have to find the volume of wood in entire stand.}$

Means we have to find Volume of cuboid - Volume of three conical depressions.

$Volume\:of\:cuboid\:stand$ = l × b × h

= ${15\:\times\:10\:\times\:3.5}$

= $525{cm}^{3}$

$Volume\:of\:3\:conical\:depressions$ = $3$×$\frac{1}{3}$× π${r}^{2}h$

= $3$ × $\frac{1}{3}$ × $\frac{22}{7}$ × ${(0.5)}^{2}$ × $1.4$

= $3$ × $\frac{1}{3}$ × $\frac{22}{7}$ × $0.25$ × $1.4$

= $1.1\:{cm}^{3}$

Now,

$\textbf{Volume of wood in the entire stand}$ = $\textbf{Volume of cuboid}$ - $\textbf{Volume of three conical depressions.}$

= $\textbf{525 - 1.1}$

= 523.9 cm³

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