Solving PV & Tangent Line Equations in Calculus

[cscott]

PV = C (Boyle's Law)

At a certain instant, the volume is 480 cm^3, the pressure is 160 kPa, and the pressure is increasing at a rate of 15 kPa/min. At what rate is the volume decreasing at this instant?

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Find the equations of both lines that pass through the point (2, 3) and are tangent to the parabola y = x^2 + x.



I don't really know where to start with these...

 

[benorin]

PV=C\Rightarrow\frac{d}{dt}(PV)=\frac{d}{dt}C\Rightarrow \frac{dP}{dt}V+P\frac{dV}{dt}=0

then plug-in the known values of P,V, and dP/dt to solve for dV/dt.

 

[HallsofIvy]

Any line through (2, 3) can be written as y= m(x- 2)+ 3 for some m.

If (x,y) is a point where that line intersects the parabola y= x²+ x, then we must have m(x-2)+ 3= x²+ x. If, in addition, the line is tangent to the parabola there, we must have
m= 2x+ 1. Solve those two equations for x and m.

I've edited this: before I had m= 2x- 1. Obviously, the derivative of x²+ x is 2x+ 1, not 2x- 1.

 

[cscott]

Thank you.