Jessica and Matthew are running toward the point P along the straight paths that

Question

Jessica and Matthew are running toward the point P along the straight paths that make a fixed angle of theta (Figure 1). Suppose that Matthew runs with velocity v_a (m/s) and Jessica with velocity v_b (m/s). Let f(x, y) be the distance from Matthew to Jessica when Matthew is x meters from P and Jessica is y meters from P Show that f(x, y) = squareroot x^2 + y^2 - 2xycos theta. Assume that theta = pi/3. Use the Chain Rule to determine the rate at which the distance between Matthew and Jessica is changing when x = 10, y = 27, v_a = 4 m/s, and v_b = 3 m/s.

Explanation / Answer

Solution:

(a) This is a simple application of the Law of Cosines.

Connect points A and B in the diagram to form a line segment that we will call f .

Then, the Law of Cosines says that f 2 = x2 + y2 - 2xy cos theta

. By taking square roots, we find that f =sqrt (x2 + y2 -2xy cos theta)

. (b) Using the chain rule, so we get

df /dt =((.x - y cos theta)dx/dt )/(sqrt(x2 + y2 -2xy cos theta)) + .((y -x cos theta)dy/dt) /sqrt (x2 + y2 - 2xycostheta ) and

using x = 10, y = 27, and dx/dt = 4, dy/dt = 3, we get cospi/3=0.5

df /dt =((10-27(0.5)4))/sqrt((10)2+27)2 -2(10)(27)0.5)+((27-10(0.5)(3))/(sqrt(102 +272 -2(10)(27)(0.5))

=52/sqrt(559)=2.19

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