A hydroelectric dam generates electricity by forcing water through turbines. Sediment accumulating behind the dam, however, will reduce the flow and eventually require dredging. Let y(t) be the amount of sediment (in thousands of tons) accumulated in t years. If sediment flows in from the river at the constant rate of 80 thousand tons annually, but each year 10% of the accumulated sediment passes through the turbines, then the amount of sediment remaining satisfies the differential equation y' = 80 − 0.1y. (a) By factoring the right-hand side, write this differential equation in the form y ' = a(M − y). Note the value of M, the maximum amount of sediment that will accumulate. (Enter y' as yp.) (b) Solve this (factored) differential equation together with the initial condition y(0) = 0 (no sediment until the dam was built). (c) Use your solution to find when the accumulated sediment will reach 85% of the value of M found in step (a). This is when dredging is required. (Round your answer to the nearest year.)
Solve by recognizing that the differential equation is one of the three types whose solutions we know.
A hydroelectric dam generates electricity by forcing water through turbines. Sediment accumulating behind the dam, however, will reduce the flow and eventually require dredging. Let y(t) be the amount of sediment (in thousands of tons) accumulated in t years. If sediment flows in from the river at the constant rate of 80 thousand tons annually, but each year 10% of the accumulated sediment passes through the turbines, then the amount of sediment remaining satisfies the differential equation
(a) By factoring the right-hand side, write this differential equation in the form y ' = a(M − y). Note the value of M, the maximum amount of sediment that will accumulate. (Enter y' as yp.)
(b) Solve this (factored) differential equation together with the initial condition y(0) = 0 (no sediment until the dam was built).
(c) Use your solution to find when the accumulated sediment will reach 85% of the value of M found in step (a). This is when dredging is required. (Round your answer to the nearest year.)
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