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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 54
Capitolo 3, Problema 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?

Guida verificata passo dopo passo
1
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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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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Percorso guidato
04:16
Intro To Related Rates

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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Percorso guidato
05:14
Finding The Implicit Derivative

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.
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