Quest (70 pts.): Improper Integrals, Min Path, Motion, Optimization

Honors AP Calculus / Mr. Hansen	Name: _______________________________________
3/8/2007

Calculator is OK throughout, but show adequate work.
Each problem is worth 14 points.
This sheet will be yours to keep. Submit all graded work on your own paper.

1.	Compute , showing FTC and all work.

2.	Mr. Hansen’s Theory of Boredom (TOB) states that the boredom that a student experiences is approximately 70% explainable by EB (explicable boredom), while the other 30% (inexplicable boredom, or IB) is due to random factors. EB is computed as the distance of the point (a, p) from the origin, where x = a is the student’s alertness on a scale of –10 (completely sleepy) to 10 (completely caffeinated). Clearly, a = 0 is ideal. Similarly, y = p indicates the pacing of the class relative to the student’s emotional commitment: –10 is way too slow, 10 is way too fast, and 0 is just right. A student who is slightly caffeinated (a = 3) is starting to lose his caffeine buzz at a rate of 0.5 units per minute, and the pacing of the class is constant at p = –2 (slightly too slow). Compute the rate of change of the student’s EB.

3.	St. Albam’s Monastery School (SAMS) is located on the beautiful Sams River in Leicesterworcestershire, England. The Lower School and the Upper School are on opposite banks of the river. The river is straight and 800 m wide in this vicinity, and the Upper School is situated 550 m downstream from the Lower School. The IT staff of SAMS needs to conduct high-speed Internet and data transfer between the two divisions of the school and has proposed a high-bandwidth fiber optic cable to serve as the link. Unfortunately, armored underwater cable is extremely expensive (£700 per meter), and even the land cable is expensive (£300 per meter) because of the need to bury it and protect it from moles, groundhogs, wolves, other carnivores, and curious students. Compute the optimal geometry for the water crossing. In other words, at what point should the underwater cable connect to the land cable? BONUS (1 pt.): Estimate the cost of the optimal solution to the IT staff’s requirement.

4.	An artillery shell shot from a cannon from a platform at height 10 m follows the trajectory given by y = 10 + 200t – 4.9t², where y = height (in m) above ground, t = elapsed time in seconds, and x = 200t = horizontal position (in m). Compute dy/dt, dx/dt, the net vertical displacement in the first 40 seconds of flight, and the total vertical distance traveled by the shell in the first 40 seconds of flight.

5.	A cylindrical aluminum beverage can is to be manufactured so that its volume is 350 ml, i.e., 350 cubic centimeters. Subject to the constraint , where r is in centimeters, find the dimensions (radius and height) that (a) minimize the total surface area, and (b) maximize the total surface area.