A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into cylindrical shaped small bottles of diameter 3 cm and height 4 cm. How many bottles will be needed to empty the bowl?

1. 27

2. 35

3. 54

4. 63

Given,

Internal radius of hemispherical bowl, R = 9 cm

Radius of cylindrical bottles, r = $\dfrac{\text{diameter}}{2} = \dfrac{3}{2}$ = 1.5 cm

Height of the cylindrical bottles, h = 4 cm

Let number of cylindrical bottles needed be n.

∴ Volume of hemispherical bowl = n × Volume of each cylindrical bottle

$\Rightarrow \dfrac{2}{3}π\text{R}^3 = \text{n} \times π\text{r}^2\text{h} \\[1em] \Rightarrow \dfrac{2}{3}\text{R}^3 = \text{n} \times \text{r}^2\text{h} \\[1em] \Rightarrow \dfrac{2}{3} \times 9^3 = \text{n} \times 1.5^2 \times 4 \\[1em] \Rightarrow \dfrac{2}{3} \times 729 = \text{n} \times 2.25 \times 4 \\[1em] \Rightarrow 2 \times 243 = \text{n} \times 9 \\[1em] \Rightarrow \text{n} = \dfrac{486}{9} \\[1em] \Rightarrow \text{n} = 54.$

Hence, option 3 is the correct option.