If water is being pumped into the trough at 2 cubic feet per minute, then the water level rising (in feet per minute) when h is 1.6 foot deep is 0.1 feet cubic per minute.
Suppose that we have a function y = f(x), and wish to find the rate of change of y with respect to a third variable t, where we do not have y expressed explicitly in terms of t. We use the chain rule in differentiation to find this rate of change in a process called implicit differentiation.
dy/dt = dy/dx × dx/dt
If base is equal to the height, by similar triangles:
Volume v = base x height x length/2
= bhl/2
= h²12/2
= 6h².
Therefore,
dv/dt = 12hdh/dt
= 2 ft³/min
Now,
dh/dt = 2/(12h)
when h = 1.6,
dh/dt = 2/(12 × 1.6) ≈ 0.1 ft³/min
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