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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 54
Chapter 3, Problem 54

Robert Boyle (1627–1691) found that for a given quantity of gas at a constant temperature, the pressure P (in kPa) and volume V of the gas (in m³) are accurately approximated by the equation V=k/P, where k>0 is constant. Suppose the volume of an expanding gas is increasing at a rate of 0.15 m³/min when the volume V=0.5 m³ and the pressure is P=50 kPa. At what rate is pressure changing at this moment?

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Start by understanding Boyle's Law, which states that for a given quantity of gas at constant temperature, the product of pressure and volume is constant: V = k/P. Here, k is a constant.
Given that the volume V is increasing at a rate of 0.15 m³/min, we need to find the rate at which the pressure P is changing. This involves differentiating the equation V = k/P with respect to time t.
Differentiate both sides of the equation V = k/P with respect to time t. Use the chain rule for differentiation: dV/dt = -k/P² * dP/dt.
Substitute the known values into the differentiated equation. You have dV/dt = 0.15 m³/min, V = 0.5 m³, and P = 50 kPa. Also, find the constant k using the equation V = k/P, which gives k = V * P.
Solve for dP/dt, the rate of change of pressure, using the substituted values in the differentiated equation. Rearrange the equation to isolate dP/dt and calculate its value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Related Rates

Related rates involve finding the rate at which one quantity changes in relation to another. In this problem, we need to determine how the pressure of the gas changes as its volume changes. This requires applying the chain rule of differentiation to relate the rates of change of volume and pressure.
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Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the variables are not isolated. In this case, we will differentiate the equation V = k/P with respect to time to find the relationship between the rates of change of volume and pressure. This allows us to solve for the unknown rate of change of pressure.
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Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the product of pressure and volume is a constant (PV = k). This relationship is crucial for understanding how changes in volume affect pressure. In this problem, we use the equation V = k/P to analyze the behavior of the gas as its volume increases.
Related Practice
Textbook Question

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a. At what time is the rate of change of the charge Q' the greatest?