The relationship between the pressure ppp and temperature ttt of a specific gas model is described by the equation
p3t+2pt3=12 p^3t + 2pt^3 = 12 p3t+2pt3=12Prove that the curve representing this relationship does not intersect the coordinate axes in the ppp-ttt plane.
Show that
dtdp=−3p2t+2t3p3+6pt2 \frac{dt}{dp} = -\frac{3p^2t + 2t^3}{p^3 + 6pt^2} dpdt=−p3+6pt23p2t+2t3Prove that the model has no stationary points.
In the case when p>0p > 0p>0, find the equation of the tangent line to the curve at the point where t=1t = 1t=1. Give your answer in the form at+bp=cat + bp = cat+bp=c, where a,b,ca, b, ca,b,c are integers.
