Chapter 6.1, Problem 32E

Bacteria on a Petri Dish. A petri dish is a small, shallow dish of thin glass or plastic, used especially for cultures in bacteriology. A 2-inch-radius petri dish, containing nutrients upon which bacteria can multiply, is smeared with a uniform suspension of bacteria. Subsequently, spots indicating colonies of bacteria appear. The distance of the center of the first spot to appear from the center of the petri dish is a variable with density curve y = x/2 for 0 < x < 2, and y = 0 otherwise.

1. a. Graph the density curve of this variable.
2. b. Show that the area under this density curve to the left of any number x between 0 and 2 equals x²/4.

What percentage of the time is the distance of the center of the first spot to appear from the center of the petri dish

1. c. at most 1 inch?
2. d. between 0.25 inch and 1.5 inches?
3. e. more than 0.75 inch?