A deep-sea research submersible's internal cabin pressure, P, in atmospheres (\text{atm}),… — SAT

A deep-sea research submersible's internal cabin pressure, $P$ , in atmospheres ( $\text{atm}$ ), is modeled by a linear function of its depth below the ocean surface, $d$ , in meters. At the surface ( $d = 0$ ), the internal pressure is $1.0\text{ atm}$ . For every increase in depth of $100\text{ meters}$ , the internal pressure increases by $0.05\text{ atm}$ . The submersible descends from the surface at a constant rate of $2.5\text{ meters per second}$ . What is the rate of increase of the internal cabin pressure, in $\text{atm}$ per hour, as the submersible descends?

Answer: 4.5 atm per hour

Answer

The rate of increase of the internal cabin pressure is

4.5

atmospheres per hour.

The rate of change of internal pressure with depth is

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

. Since the submersible descends at a rate of

2.5\text{ m/s}

, we convert this rate of depth change to hours:

2.5\text{ m/s} \times 3600\text{ s/hr} = 9000\text{ m/hr}

. Multiplying the pressure change per meter by the depth change per hour gives the rate of change of the internal pressure per hour:

0.0005\text{ atm/m} \times 9000\text{ m/hr} = 4.5\text{ atm/hr}

Step-by-Step Solution

Calculate the rate of pressure increase per meter of depth.

0.0005\text{ atm/m}

The pressure increases by

0.05\text{ atm}

for every

100\text{ meters}

of depth, giving a rate of change of

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

Determine the distance descended by the submersible in one hour.

9000\text{ meters}

With

3600\text{ seconds}

in one hour and a descent speed of

2.5\text{ m/s}

, the submersible descends a total of

2.5\text{ m/s} \times 3600\text{ s} = 9000\text{ meters}

in one hour.

Calculate the rate of internal pressure increase per hour.

4.5\text{ atm/hr}

Multiply the rate of change of pressure per meter (

0.0005\text{ atm/m}

) by the hourly descent distance (

9000\text{ m}

) to obtain

0.0005 \times 9000 = 4.5\text{ atm/hr}

Key Concept

Interpreting Linear Relationships in Context

Alternative Method

Write the cabin pressure as a function of time

t

, in seconds:

P(t) = 1.0 + 0.0005(2.5t) = 1.0 + 0.00125t

. The slope of this line,

0.00125\text{ atm/s}

, represents the rate of increase per second. To convert this rate to hours, multiply by

3600\text{ seconds/hour}

0.00125 \times 3600 = 4.5\text{ atm/hr}

Estimated Time:2m 0s

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Answer

Step-by-Step Solution

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