Inspired by Dish-washing!

Consider the following diagram-

L is a murky ideal liquid of refractive index $$n=2.23$$ filled in a container C of cross section Z to a height of H= 64cm. Container C has an orifice O (near the bottom) of cross section z from which the liquid flows out starting at a time t = 0. L has a unique property that the intensity of light transmitted through it varies as $$I=I_{initial}e^{-10d}$$ where d is the distance travelled by light through the the liquid.The bottom R of C is reflecting (perfectly). A (of mass 50g) consists of a floating light emitter (it emits a narrow beam of light) a and receiver b which lie very close to each other. a emits light of intensity $$I_{0}$$ and $$b$$ responds only when it receives light of intensity greater than $$0.04076I_{0}$$. If t(to the nearest positive integer) is the time after which b responds and if U is the loss in potential energy of A in the time interval between the emission of the light ray which produces the first response and the reception of the same such that $$U=m10^{-x}$$,

find |t| + x.

Given: - $$\frac{z}{Z} =10^{-4}$$

• $$c= 3 \times 10^{8}ms^{-1}$$

• $$g= 10ms^{-2}$$

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