(As opposed to all the bullshit textbook "real world" problems)

From an article in American Mathematical Monthly from a long time ago. All should be solvable with first-semester calculus.

Problem 1) Find the centroid of Nevada.  (For simplicity you may assume Nevada is a perfect trapezoid lying in a plane, and that one degree of lattitude or longitude is a constant distance throughout Nevada.)

Problem 2) When crude oil flows from a well, water is frequently mixed with it in an emulsion.  To remove the water the crude is piped to a device called a heater-treater, which is simply a large tank in which the oil is warmed and the water is allowed to settle out.  Operating experience in a particular oil field indicates that the concentration $C$ of water in the treater's output can be modeled by the following equation in a neighborhood of the usual operating point of 135°F and a 2-hour holding time:

$C = 0.04 - 0.0032h^2 - 10^{-6} t^2 + 4.1 \cdot 10^{-5} ht$,

where $h$ is the holding time in hours and $t$ is the operating temperature in degrees F.  (a) Because of random fluctuations in the well's flow rate, the holding time actually varies slightly around 2 hours.  Suppose you are given a simple control device that can change the tank temperature proportionally to the measured change in holding time.  What constant of proportionality would best compensate for small holding time fluctuations and keep the water concentration as constant as possible? (b)  Now as field equipment ages, its maximum operating settings are generally decreased.  Find the equation of the line that best approximates the way in which the holding time would have to be increased as the maximum temperature rating falls slightly below the usual operating temperature.  (This is a problem an engineering student was asked to solve while working for an oil company one summer).

Problem 3) Your friend is at the top of a building which you know to be exactly 100 feet high.  She throws a stone downward at $v_0$ ft/sec, and you time how long it takes to hit the ground.  Neglect air resistance, so that the distance (in feet) fallen by the stone in $t$ seconds is given by the formula $s=16t^2 + v_0 t$.  If you time 2 seconds for the stone to fall to the ground, and if your stopwatch is accurate only to within $\pm 0.1$ sec, at what downward speed did your friend throw the stone?  Include an error estimate in your answer.  Use the tangent line approximation.

Problem 3A) If an object falls vertically from rest with air resistance proportional to velocity, then its velocity at time $t$ is given by the formula $v(t) = \frac{g}{k} \left(1-e^{-kt}\right)$.  Here $g = -9.8 m/sec^2$ is known exactly, and $k$, the constant of proportionality between the air resistance and the velocity, is measured experimentally with some error $\Delta k$.  (a) Derive a formula for the height $s(t)$ in terms of the initial height $s_0$ and $g,k,t$. (b) Suppose you measure that it takes 2.0 seconds for the object to hit the ground, with an error of $\Delta t$ in this measurement.  Your wind tunnel experiments give a value of 0.1 for $k$, with an error of $\Delta k$.  Find a simple formula for $s_0$ in terms of $\Delta k$ and $\Delta t$.  Use the tangent plane approximation.

Problem 4) You are standing on the ground at a point $B$ (see diagram: http://i29.tinypic.com/4j2tds.jpg ), a distance of 75 feet from the bottom of a ferris wheel that is 20 feet in radius.  Your arm is at the same level as the bottom of the ferris wheel.  Your friend is on the ferris wheel, which makes one revolution (counterclockwise) every 12 seconds.  At the instant when she is at point $A$ you throw a ball to her at 60 ft/sec at an angle of 60° above the horizontal.  Take $g=-32 ft/sec^2$, and neglect air resistance.  Find the closest distance the ball gets to your friend, using Newton's method to obtain an answer which is accurate to within 1/2 foot.