Motion System Design

# Fun with Fundamentals: Problem 255

Problem 255: Battle hymn bump
Glory, glory awaits if you solve this problem.

Tragedy struck last week. Specifically, it struck teacher Mrs. Ironfiss in the head, in the form of a tiny, rotten tangerine. And now her head is chartreuse. It seems that one of her students, after being reprimanded with a ruler, ambushed her after school with an overripe fruit as a form of retribution. Working from a second story window, the culprit lobbed the projectile 200 feet, intersecting the instructorâ€™s noggin, or bean, if you will, just as she was approaching her car, a 1987 light blue Buick Riviera with sticky power windows. As she watched in her rearview mirror, the resulting symmetrical protrusion grew to an amazing ten millimeters high.

Fortunately for the student, no one witnessed the act. Unfortunately, Mrs. Ironfiss has a knack for making everything an educational experience:

the green bump left on my head by the tangerine is receding daily now at a constant rate, with a cross-section that can be approximated as one full period of y = 5cos(0.25x), gradating into unswollen noggin at the troughs. Assuming my spherical head has a diameter of 180 mm, what is the volume of the bump?

Solution to problem 254, February 2002: Chocolate challenge