For a certain gas

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For a certain gas, pressure PP (in kPa) depends on volume VV (in L) by a differentiable, invertible function P=q(V)P=q(V). At V=2V=2 L, P=100P=100 kPa and dPdV=−40\dfrac{dP}{dV}=-40 kPa/L. Volume in turn depends on temperature θ\theta (in ∘ ⁣C^{\circ}\!\text{C}) through a differentiable, invertible function V=r(θ)V=r(\theta) with r(300)=2r(300)=2 and dVdθ=0.01\dfrac{dV}{d\theta}=0.01 L/°C at θ=300 ∘ ⁣C\theta=300\,^{\circ}\!\text{C}. What is dθdP\dfrac{d\theta}{dP} at P=100P=100 kPa? (a) −0.025  ∘ ⁣C ⁣/ ⁣kPa-0.025\;^{\circ}\!\text{C}\!/\!\text{kPa} (b) −0.4  ∘ ⁣C ⁣/ ⁣kPa-0.4\;^{\circ}\!\text{C}\!/\!\text{kPa} (c) −2.5  ∘ ⁣C ⁣/ ⁣kPa-2.5\;^{\circ}\!\text{C}\!/\!\text{kPa} (d) 2.5  ∘ ⁣C ⁣/ ⁣kPa2.5\;^{\circ}\!\text{C}\!/\!\text{kPa}

Related theory

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