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

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

Step-by-step 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*(1ft)/(12in)=318.75 ft^(3)*7.48(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=(V_(0))/((r_(out)-r_(in))) =(2384.25 gal)/((73.8-57.5)gal/min)=`146,27 min