A cylindrical vessel of \(40\) cm radius is completely filled with water and its capacity is \(528\) dm\(^3\) (dm : decimeter). The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at \(70\) cm below the top of water level, then horizontal range of water falling on the ground in the beginning is __________ cm.

Volume of the cylindrical vessel, \(V = 528\) dm\(^3 = 528000\) cm\(^3\)

Radius of the vessel, \(r = 40\) cm

Let the height of the vessel be \(H\).

\(V = \pi r^2 H\)

\(528000 = \dfrac{22}{7} \times (40)^2 \times H\)

\(H = \dfrac{528000 \times 7}{22 \times 1600} = 105\) cm

The vessel is placed on a block of height equal to the vessel, so the height of the block is \(105\) cm.

Depth of the hole from the top water level, \(h = 70\) cm

Height of the hole from the ground, \(y = \text{Height of block} + (H - h)\)

\(y = 105 + (105 - 70) = 140\) cm

Velocity of efflux, \(v = \sqrt{2gh}\)

Time of flight of water, \(t = \sqrt{\dfrac{2y}{g}}\)

Horizontal range, \(R = v \times t = \sqrt{2gh} \times \sqrt{\dfrac{2y}{g}} = 2\sqrt{hy}\)

\(R = 2\sqrt{70 \times 140} = 2\sqrt{70 \times 70 \times 2} = 140\sqrt{2}\) cm

Answer: \(140\sqrt{2}\)