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(1) A snowplow's velocity v(t) depends on the depth D(t) of the snow that it is plowing. We'll approximate this relationship as v ( t ) = k 1 + D ( t ) where D(t) is the depth of the snow at time t, and k is the maximum speed of the snowplow when there is no snow at all. Some time during the night it begins to snow at a constant rate, and continues to snow at this rate indefinitely. At 7am a snowplow begins to plow, driving in a straight line. By 8am, the snowplow has travelled 22 miles, by 9am 28 miles and by 10am 33 miles. Your goal is to figure out what time it started snowing. (a) Write an equation for the depth of the snow at time t, where t measures hours after 7am. (There should be 2 unknown constants in your equation, r the rate of snowfall, and d, the depth at time t = 0.) (b) Write a differential equation for the position x of the plow (in miles from the garage) at time t. (The constants k, r, and d should appear in the equation.) (c) Solve this differential equation. Then use the initial conditions to eliminate the vari- ables r and d, this should leave an equation involving only k. While difficult to solve, you can solve it numerically (for example, use the command find root (f (x) ==0, a, b) in SageMath to find a root of f(x) in the interval [a, b]. Take the value of k you find and round it to the nearest integer for the remainder of this problem. (d) Now, solve for d and r, give their exact values to 4 decimal places. What time did it begin snowing? (2) A second snowplow (identical to the first) departs at 8am and follows in the first snow- plow's tracks. (a) Assume that immediately after the first snowplow passes, the depth of the snow is 0. Write a function D2(y, t) for the depth of the snow at time t and at distance y along the road. (Assuming the first snowplow has already passed.) (b) Write a differential equation for the velocity of the second snowplow as a fuction of time t and position y along the road. Then use this to find a differerential equation for the change in time as a function of the position y along the road (dt). (Note: this is the reciprocal of velocity! Hint: writing it this way should give a linear differential equation of t as a function of y.) c) Solve this differential equation. (d) At what time does the second snowplow crash into the back of the first? (You'll have to solve an equation numerically again...)