f(x,y) chain rule problem Jessica and Matthew are running toward the point P alo
ID: 2850136 • Letter: F
Question
f(x,y) chain rule problem
Jessica and Matthew are running toward the point P along the straight paths that make a fixed angle of O (Figure 1). Suppose that Matthew runs with speed va (m/s) and Jessica with speed vb (m/s). Let f(x, y) be the distance from Matthew to Jessica when Matthew is z meters from P and Jessica is y meters from P. FIGURE 1 Show that f(x, y) = root x^2 + y^2 - 2xy costheta Assume that theta = pi/3. Use the Chain Rule to determine the rate at which the distance between Matthew and Jessica is changing when Note: Be careful about the signs of dx/dt and dy/dt. Use the information provided to work out whether each is positive or negative.Explanation / Answer
in triangle APB let third side be z
by law of cosines ,
z2=x2+y2-2xycos
==>z=[x2+y2-2xycos]
f(x,y)=[x2+y2-2xycos]
=pi/3
f(x,y)=[x2+y2-2xycos(pi/3)]
f(x,y)=[x2+y2-xy]
differentiate with respect to t
df/dt =[1/2[x2+y2-xy]]*(2xdx/dt +2ydy/dt -ydx/dt -xdy/dt]
df/dt =[1/2[232+272-23*27]]*(2*23*(-4) +2*27*(-4) -27*(-4) -23(-4) ]
df/dt =[1/2[232+272-23*27]]*(2*23*(-4) +2*27*(-4) -27*(-4) -23(-4) ]
df/dt =-3.962
distance decreasing at 3.962 m/s
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