Solving Related Rates Problems

Big idea: use chain rule to compute rate of change of distance between two vehicles.Plan: Choose coordinate system: Let the y-axis point North and the x-axis point East.

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In this case, we say that \(\frac\) and \(\frac\) are related rates because \(V\) is related to r.Because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields.Differentiation with respect to time or one of the other variables requires application of the chain rule, Errors in this procedure are often caused by plugging in the known values for the variables before (rather than after) finding the derivative with respect to time.For example, one can consider the kinematics problem where one vehicle is heading West toward an intersection at 80 miles per hour while another is heading North away from the intersection at 60 miles per hour.One can ask whether the vehicles are getting closer or further apart and at what rate at the moment when the North bound vehicle is 3 miles North of the intersection and the West bound vehicle is 4 miles East of the intersection.Therefore, \(t\) seconds after beginning to fill the balloon with air, the volume of air in the balloon is \(V(t)=\fracπ[r(t)]^3cm^3.\) Differentiating both sides of this equation with respect to time and applying the chain rule, we see that the rate of change in the volume is related to the rate of change in the radius by the equation \(V'(t)=4π[r(t)]^2r′(t).\) The balloon is being filled with air at the constant rate of 2 cm3/sec, so \(V'(t)=2cm^3/sec.\) Therefore, \(2cm^3/sec=(4π[r(t)]^2cm^2)⋅(r'(t)cm/s),\) which implies \(r'(t)=\fraccm/sec\).When the radius \(r=3cm,\) \(r'(t)=\fraccm/sec.\) Note that when solving a related-rates problem, it is crucial not to substitute known values too soon.One vehicle is headed North and currently located at (0,3); the other vehicle is headed West and currently located at (4,0).The chain rule can be used to find whether they are getting closer or further apart.Interpreting the Problem Setting up the Solution Solving a Sample Problem Involving Triangles Solving a Sample Problem Involving a Cylinder Show 1 more... Questions & Answers Related Articles References This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness.Together, they cited information from 7 references.