Question

ly the techniques discussed in this chapter to real-world problems, it is necessary o late these problems into questions that can be answered mathematically lating a real-world problem as a mathematical one often requires assumptions. To illustrate this, consider the following snowplow problem: The process o f reformu- making certain simplifying One morning it began to snow very hard and continued snowing steadily throughout the day. A snowplow set out at 9:00 A.M. to clear a road, clearing 2 mi by 11:00 A.M. and an additional mile by 1:00 P.M. At what time did it start snowing? To solve this problem, you can make two physical assumptions concerning the rate at which it is snowing and the rate at which the snowplow can clear the road. Because it is snowing steadily, it is reasonable to assume it is snowing at a constant rate. From the data given (and from our experience), the deeper the snow, the slower the snowplow moves. With this in mind, assume that the rate (in mph) at which a snowplow can clear a road is inversely proportionalto the eli ely proportional to the depth of the snow.

Answer 1

show plow set out at 9: 00 AM t = 0 the function for snow a ds/dt = c ds = c dt s = ct + D, c notequalto 0 s = Rate of snow then the rate of snow to clean the road dy/dt = k/s dy/dt = k/ct + D dy = k/ct + D dt y = k ln(ct + D)/c + e therefore y(t) = k/c ln (ct + D) + epsilon \r\n initial condition is Y(t) = 0 at 9: 00 Am y(0) = k/c ln (c(0) + D) + epsilon 0 = k/c ln (0) + 0 since k notequalto 0, ln D = 0 D = 1 initial conditions at 11: 00 AM is t = 2 then y(2) = k/c on (2c + 1) y(2) = 2m: at 11: 00 Am cleared k/c ln(2l + 1) = 2m initial conditions at 11: 0 am is t = 4 y(4) = k/c ln(4c + 1) y(4) = k/c ln(4c + 1)