due Friday, Jan. 25
Note: An entry such as ``2.1.6'' means question 6 in section 2.1 of Boyce and DiPrima.You will find "Newton's Applet" useful for solving equations numerically when no algebraic solution is available.
2.2.18 Solution
2.2.22 Solution
2.3.5 Solution
2.4.15 Solution
2.5.4 Solution
2.5.16 Solution
Note: Critical points are the same as equilibrium points, and at this point ``asymptotically stable'' is the same as ``stable'' (the distinction won't come up until Chapter 7).
E.1 A small lake contains 105 cubic metres of water. Stream A flows into the lake at a rate of 100 cubic metres per day, and stream B flows out of the lake at the same rate (rainfall, evaporation etc. are assumed to cancel each other out, so the lake volume stays constant). The water in the lake is originally pure, but at time t=0 a factory begins to dump a certain pollutant into Stream A at an increasing rate, so that the concentration of pollutant in this stream at time t days is 3 t milligrams per cubic metre. The pollutant can only leave the lake by flowing out Stream B. We assume that the lake water is well-mixed, so the concentration of pollutant in Stream B is the same as the concentration in the lake.
(a) What is the concentration of pollutant in the lake at time t days?
(b) Approximately how long does it take until the concentration of pollutant in the lake is 20000 milligrams per cubic metre? Hint: if t is large, one of the terms in the solution can be neglected.
E.2 A skydiver with a mass of 75 kg (including equipment) falls vertically from rest. The force of air resistance is -10 v Newtons when the parachute is not open and -k v Newtons when it is open, where v is the velocity in metres per second and k is a constant.
(a) What is the skydiver's terminal velocity when the parachute is not open?
(b) The skydiver opens the parachute when she reaches 90% of the terminal velocity from (a). When does this happen, and how far has she fallen by this time?
(c) The skydiver hits the ground 10 seconds after opening her parachute, at a speed of 4 metres/second. This is very close to the terminal velocity with the parachute open. What is the total distance she falls?
E.3 My bathtub has a flat bottom and vertical sides, so the volume of water in it is proportional to the depth. When water drains out, it does so at a rate proportional to the square root of the depth (this is because, in a perfect fluid, the kinetic energy of the water as it drains out should be equal to the drop in potential energy from the water surface to the drain). I start with an empty tub, stop up the drain and turn on the tap. It takes 10 minutes for the tub to fill completely. Then I turn off the tap, remove the stopper and the tub drains completely in 8 minutes. Now I turn on the tap again, but leave the stopper out. How full will the tub get?