Question

A wooden toy was made by scooping out a hemisphere of same radius from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the volume of wood in the toy. $\left[ Use\ \pi \ =\ \dfrac{22}{7} \right]$
A. 425
B. 290
C. 474.83
D. 205.33

Answer 1

Hint: Volume of cylinder and hemisphere are given by relation $\pi {{r}^{2}}h$ and $\dfrac{2}{3}\pi {{r}^{3}}$ where r is the radius, h is the height of cylinder and $\pi \ =\ \dfrac{22}{7}$.Volume of the toy is the difference of the volume of the toy and two hemispheres on the basis of the cylinder.

Complete step-by-step answer:
As we have given that a toy is made by scooping out a hemisphere of same radius from each end of the solid cylinder. It means we are cutting two hemispheres from both the ends of the cylinder.
Hence, we can draw diagram as

Now we can get the volume of the toy by taking the difference of volume of the cylinder and volume of both the hemispheres. So, we get
Volume of the toy = Volume of cylinder – 2 $\times$ Volume of one hemisphere ……………………………(i)
Here, we have abstracted 2 $\times$ Volume of one hemisphere from the volume of the cylinder in equation (i) because volume of both the hemispheres will be equal by symmetry.
Now, we know the volume of the cylinder and hemisphere can be given as $\pi {{r}^{2}}h$ and $\dfrac{2}{3}\pi {{r}^{3}}$ respectively where ‘r’ is the radius and ‘h’ is the height of the cylinder.
So, volume of cylinder in toy can be given as
Volume of cylinder in toy $=\ \pi {{r}^{2}}h$
Where $\pi \ =\ \dfrac{22}{7}$, r = 3.5 cm and h = 10 cm
Hence, we get
Volume of cylinder in toy $=\ \dfrac{22}{7}\times {{\left( 3.5 \right)}^{2}}\times 10$
$=\ \dfrac{22}{7}\times 3.5\times 3.5\times 10$
$=\ 22\times 0.5\times 3.5\times 10$
$=385\ c{{m}^{3}}$
Now, volume of the hemisphere can be given by
Volume of one hemisphere in toy $=\ \dfrac{2}{3}\pi {{r}^{3}}$
We know that the radius of the hemisphere is equal to the radius of the cylinder as both are on the same base.
Hence, $r=3.5cm$ and $\pi \ =\ \dfrac{22}{7}$ $r=3.5cm$
So, volume of one hemisphere in Toy $=\ \dfrac{2}{3}\times \dfrac{22}{7}\times {{\left( 3.5 \right)}^{3}}$
$=\ \dfrac{44}{3}\times \dfrac{3.5\times 3.5\times 3.5}{7}$
$=\ \dfrac{44}{3}\times 3.5\times 3.5\times 3.5$
$=\ \dfrac{269.5}{3}\ c{{m}^{3}}$
Now, we can put the value of volumes of hemisphere and cylinder in the equation (i) to get the value of the Toy.
Hence, we get
Volume of toy $=\ 385-\dfrac{269.5}{3}\times 2$
$=\ \dfrac{385\times 3-269.5\times 2}{3}$
$=\ \dfrac{616}{3}\ =\ 205.33$
Hence, we get volume of toy $=\ 205.33\ c{{m}^{3}}$
Hence, option (D) is the correct answer.

Note: Formulae for volumes of hemisphere and cylinder are the key points.Don’t get confused with the radius of the hemisphere, it will be equal to the radius of the cylinder as bases of both of them are the same.