# Mathematics

MAT 3613.001 Exam 1 Take-Home Part NAME

1. Solve the following problem. First, set up the di�erential equation with its initial condition, then solve. A tank contains 300 gal of brine solution in which 30 lb salt is initially dissolved. A brine solution with concentration 4 lb/gal is then allowed to �ow into the tank at a rate of 3 gal/min and the well-stirred mixture �ows out of the tank at the rate of 4 gal/min. Let y(t) = amount of salt in the tank at time t Find y(t).

2. A sailboat has been running (on a straight course) under a light wind at 1 m/sec. Suddenly the wind picks up, blowing hard enough to apply a constant force for 600 N to the sailboat. The only other force acting ton the boat is water resistance that is proportional to the velocity of the boat. If the proportionality constant for water resistance is b = 100 N-sec/m and the mass of the sailboat is 50 kg, �nd the equation of motion of the sailboat. What is the limiting velocity of the sailboat under this wind?

3. An industrial electromagnet can be modeled as an RL circuit, while it is being energized with a voltage source. If the inductance is 10 H and the wire windings contain 3 Ω of resistance, how long does it take a constant applied voltage to energize the electromagnet to within 90% of its �nal value (that is , the current equals 90% of its asymptotic value)?

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