Question

The hull of a vessel develops a leak and takes on water at a rate of 57.5 gal/min. When the leak is discovered the lower deck is already submerged to a level of 7.5 inches. At this time, a sailor turns on the bilge pump which begins to remove water at a rate of 73.8 gal/min. As an approximation, the lower deck can be modeled as a flat-bottomed container with a bottom surface area of 510 ft2 and straight vertical sides. How long will it be after the pump is turned on until the deck is clear of water?

Answer 1

Answer:

It will be around 146,27 min since the pump is turned on until the deck is clear of the water.

Explanation:

When the leak is discovered and the pump is turned on, the lower deck is already submerged and the leak is not fixed; then, in order to have the deck clear of water, the bilge pump has to remove the accumulated water ($V_{0}$) and the water that is taking on ($r_{in}*t$) through the leak. We can represent this mathematically as follow:

$V_{0} +r_{in} *t-r_{out}*t=0$  Equation 1

Where:

$V_{0}$: is the accumulated water when the leak was discovered

$r_{in}$: is the takes on rate through the leak = 57.5 gal/min

$r_{out}$: is the removing rate of the bilge pump = 73.8 gal/min

t= is the time since the pump is turned on until the deck is clear of water.

To calculate the accumulated water ($V_{0}$), we will model the lower deck as a flat-bottomed container with a bottom surface area of 510 $ft^{2}$ and straight vertical sides. Knowing that the level submerged is 7.5 inches, and performing the corresponding unit conversions, we obtain:

$V_{0}$= bottom surface area * lever submerged

$V_{0}= 510ft^{2}*7.5 in*\frac{1ft}{12in}=318.75 ft^{3}*7.48\frac{gal}{1ft^{3}}=2384.25 gal$ Equation 2

Solving equation 1 for time (t), and replacing the value obtained in equation 2, we get:

$t=\frac{V_{0}}{(r_{out}-r_{in})} =\frac{2384.25 gal}{(73.8-57.5)gal/min}=$146,27 min