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\u003e0 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?

Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 3 - Derivatives

문제 54

3장, 문제 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?

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.