In a cylindrical vessel containing liquid of density d, there are two holes in the side walls at height of h1 and h2 respectively such that range of efflux at the bottom of the vessel is same. Find the height of hole for which the range of efflux would be maximum. Ans( h1+h2)/2

Let the height of cylindrical vessel be H, the height of hole be h

We know that range of efflux R = 2[h*(H - h)]1/2

or R = 2(Hh - h2 )1/2

for maximum range differentiate R wrt h and equate the derivative to zero

dR/dh = (H - 2h)/(Hh - h2 )1/2

dR/dh = 0

(H - 2h)/(Hh - h2 )1/2 = 0

H - 2h = 0

H = 2h.

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Now h1 and h2 be heights of two holes such that range of efflux from them is same

2(Hh1 - h12 )1/2 = 2(Hh2 - h22 )1/2

Hh1 - h12 = Hh2 - h22

H (h1 - h2) = h12 - h22 = (h1 - h2)(h1 + h2)

H = h1 + h2

but H = 2h

2h = h1 + h2

h = (h1 + h2)/2