Question

A 4.0-mm-diameter hole is 1.0 m below the surface of a 2.0-m-diameter tank of water. What is the rate, in mm/min, at which the water level will initially drop if the water is not replenished? Express your answer to two significant figures and include the appropriate units.

Answer 1

Answer:

$1.06$ mm/min

Explanation:

h = height of the water level above the hole = 1 m

$d_{h}$ = diameter of hole = 4 mm = 0.004 m

$A_{h}$ = Area of cross-section of hole = $(0.25) \pi d_{h}^{2}$

$d_{t}$ = diameter of tank = 2 m

$A_{t}$ = Area of cross-section of tank = $(0.25) \pi d_{t}^{2}$

R = rate at which the water level drop

Rate at which the water level drop is given as

$R = \frac{A_{h} (\sqrt{2gh}))}{A_{t}}$

$R = \frac{((0.25) \pi d_{h}^{2}) (\sqrt{2gh}))}{((0.25) \pi d_{t}^{2})}$

$R = \frac{(d_{h}^{2}) (\sqrt{2gh}))}{(d_{t}^{2})}$

$R = \frac{(0.004)^{2} \sqrt{2(9.8)(1)}}{(2)^{2}}$

$R = 17.7\times 10^{-6}$ m/s

$R = 0.0177$ mm/sec

$R = (0.0177) (60)$ mm/min

$R = 1.06$ mm/min