A tank has a hole at its bottom. The time needed to empty the tank from level `h_(1)` to `h_(2)` will be proportional to

Options

A. `h_(1)-h_(2)`
B. `h_(1)+h_(2)`
C. `sqrt(h_1)-sqrt(h_2)`
D. `sqrt(h_1)+sqrt(h_2)`

Answer

C

Explanation

Let =(dh)/(dt)` represent the rate of descent of water level.
Let `A` and `a` represent the cross -sectional areas of the container and hole repectively
Then , `-A(dh)/(dt) = asqrt(2gh)`
`dt = -k(dh)/(sqrt(n)) dt`
or `int_(0)^(t) dt = -kint_(h_1)^(h_2) (1)/(sqrt(h))dh`
or `t=-k
|(h^(1/2+1))/(-1/2+1)|_(h_(1))^(h_2)`
or `t=-2k[sqrt(h_2)-sqrt(h_1)]`
or `t prop (sqrt(h_(1) )-sqrt(h_2))`.