Related Rates

Constraints among changing quantities

The solution procedure

Reading the geometry

The shadow template

In a related rates problem where variables x and y both depend on time t, what does differentiating the constraint x^2 + y^2 = 25 with respect to t produce?

At which step of the related rates procedure should the numerical values at the instant of interest be substituted?

A 5 meter ladder slides down a wall, the foot moving outward at dx/dt = 0.3 meters per second. When the foot is x = 3 meters out, the top is y = 4 meters high. How fast is the top moving?

A spherical balloon is inflated at dV/dt = 10 cubic centimeters per second. Using dV/dt = 4 pi r^2 dr/dt, how fast is the radius growing when r = 5 centimeters?

In a related rates solution, one quantity's rate comes out negative, say dV/dt = -5 cubic centimeters per second for a draining tank. What does the negative sign mean?

A related rates problem states dV/dt = 10 and asks for dr/dt at r = 5. Which quantity's rate is the unknown?

The volume of a cone is V = (1/3) pi r^2 h. If the radius is constant and only the height changes, dV/dt equals a constant times dh/dt. That constant is pi r^2 over what number? Answer with a number.

In the ladder problem with x^2 + y^2 = 25, when x = 3 the corresponding height y of the top equals what number? Answer with a number.

A related rates equation balances 2x dx/dt against 2y dy/dt, where x and y are in meters. What unit must dx/dt carry? Answer like meters per second.

A boat is pulled by a rope of length L toward a dock, with horizontal distance x from the dock and the rope arriving from a point 8 meters above the water. The constraint is x^2 + 64 = L^2. Differentiating gives 2x dx/dt equal to what expression in L and dL/dt? Answer like 2L dL/dt.