Word Problem Review This handout gives you some experience in solving word problems. This is important, because as engineers and scientists, your mathematics will be worth nothing if you cannot look at a situation from your field and see what the relevant mathematics is for the situation. In each problem read the problem over and ask, “What’s the question?” “Have I seen a problem like this before?” If you have done all of the syllabus problems, the computer labs, and looked over the back exams and review materials, then every problem on the final will be like some problem you have seen before. Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 1) A charge distribution on a plane is creating an electric field. The electrical potential P ( x, y ) measures the potential energy of a unit point charge due to its position in the field. The function is given by P ( x, y ) = 2 √ ( x +2) 2 +( y - 1) 2 a) What is the gradient of the potential at (1 , 2)? b) What is the rate of change of P in the direction ~u = - 1 2 ~ i + √ 3 2 ~ j at (1 , 2)? c) An equipotential line is a curve on our plate along which the potential is constant. What is an equation for the tangent line of the equipotential passing through (1 , 2)? d) The electric field vector is orthogonal to the equipotential lines, and points in the direction of decreasing potential. What direction does it point at (1 , 2)? (Your answer should be a unit vector.) Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 2) (10 points) A current of 34 amperes branches into currents x , y , and z through resistors with resistances 1, 4, and 6 ohms as shown. (The current I 1 through the top wire is x , while the resistor in the top wire has resistance R 1 equal to 1, etc.) It is known that the current splits in such a way that the sum of the currents through the three resistors equals the
initial current. The energy E generated in a resistor of resistance R by current I is given by E = I 2 R . It turns out that nature always splits the currents so that the total energy is minimized. Find the current in each branch. Read over the next problem, and ask yourself what kind of a problem is this, and where have I seen a similar one? (Space for your answers below) 3) The temperature z in degrees centigrade at a point on a road, x meters East and y meters North from a fixed location (the origin), is given as z = f ( x, y ). If a bug at a point on the road moves East, the temperature increases at the rate of 0 . 8 degrees/meter If it moves North, the temperature decreases at the rate of 1 . 4 degrees/meter. At time t = 0, the bug decides to move along a path ( x ( t ) , y ( t )) from its current location with velocity ( x 0 (0) , y 0 (0)) = (0 . 8 , - 1 . 4) meters/minute. What is the rate of change in temperature along the bug’s path at t = 0?
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