Two cylindrical vessels with radii 15 cm and 10 cm and heights 35 cm and 15 cm respectively are filled with water. If this water when poured into a cylindrical vessel, 15 cm in height, fills it completely then the radius of the vessel is :

1. 17.5 cm

2. 18 cm

3. 20 cm

4. 25 cm

Given,

For cylinder 1,

Radius, r = 15 cm

Height, h = 35 cm

For cylinder 2,

Radius, R = 10 cm

Height, H = 15 cm

By formula, Volume of cylinder = πr²h

Volume of cylinder 1 = v

$= \dfrac{22}{7} \times 15^2 \times 35 \\[1em] = 22 \times 225 \times 5 \\[1em] = 24750 \text{ cm}^3.$

Volume of cylinder 2 = V

$= \dfrac{22}{7} \times 10^2 \times 15 \\[1em] = \dfrac{22}{7} \times 100 \times 15 \\[1em] = \dfrac{33000}{7} \text{ cm}^3.$

Given, water from cylinder 1 and 2 is poured into cylinder 3.

Volume of cylinder 3 = Volume of cylinder 1 + Volume of cylinder 2

= 24750 + $\dfrac{33000}{7}$

= $\dfrac{173250 + 33000}{7} = \dfrac{206250}{7} \text{ cm}^3$

For cylinder 3,

Radius be a cm

Height = 15 cm

Volume of cylinder 3 = πa² × 15

$\Rightarrow \dfrac{206250}{7} = \dfrac{22}{7} \times \text{a}^2 \times 15 \\[1em] \Rightarrow \dfrac{206250}{7} \times \dfrac{7}{22 \times 15} = \text{a}^2 \\[1em] \Rightarrow \dfrac{206250}{330} = \text{a}^2 \\[1em] \Rightarrow \text{a}^2 = 625 \\[1em] \Rightarrow \text{a} = \sqrt{625} \\[1em] \Rightarrow \text{a} = 25 \text{ cm.}$

Hence, option 4 is the correct option.