A heavy-duty vehicle's pneumatic braking system relies on an accumulator tank to store pot… — Armed Services Vocational Aptitude Battery (ASVAB)

A heavy-duty vehicle's pneumatic braking system relies on an accumulator tank to store potential energy. Initially, a volume of $150 \text{ in}^3$ of air is compressed at an absolute pressure of $90 \text{ psi}$ inside the accumulator. When the operator engages the brakes, the air expands isothermally (at a constant temperature) into the connected brake chambers, causing the system's absolute pressure to drop to $30 \text{ psi}$ . What is the total volume that the air now occupies, in cubic inches?

Answer: 450 in^3

Answer

450

According to Boyle's Law, the pressure and volume of a confined gas are inversely proportional when the temperature is held constant. The initial state is

90 \text{ psi}

and

150 \text{ in}^3

. The final pressure drops to

30 \text{ psi}

, which is one-third of the original pressure. Therefore, to balance the equation and conserve the quantity of gas, the volume must expand to three times its original size.

150 \times 3 = 450 \text{ in}^3

Step-by-Step Solution

Identify the known variables from the scenario.

P_1 = 90 \text{ psi}

V_1 = 150 \text{ in}^3

P_2 = 30 \text{ psi}

Establishing the initial and final states of the gas is necessary to select the correct formula.

Select the appropriate gas law.

Use Boyle's Law:

P_1V_1 = P_2V_2

The problem states the expansion happens isothermally (constant temperature) and involves inversely proportional pressure and volume changes.

Substitute the values and solve for

V_2

90 \times 150 = 30 \times V_2

, so

V_2 = 13500 / 30 = 450 \text{ in}^3

Mathematical calculation is required to find the unknown final volume.

Key Concept

Boyle's Law (Pressure-Volume relationship at constant temperature)

Alternative Method

Instead of using the formula

P_1V_1 = P_2V_2

algebraically, you can use proportional reasoning. The pressure drops from

90 \text{ psi}

30 \text{ psi}

, which is a reduction by a factor of 3 (

90 / 30 = 3

). Because pressure and volume are inversely proportional at a constant temperature, the volume must increase by that exact same factor. Therefore, simply multiply the initial volume by 3:

150 \times 3 = 450 \text{ in}^3

Estimated Time:1m 0s

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Alternative Method

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