# Q & A: Math Homework

Q:
Actually I have 2 questions: 1. There are an average of 32 million bacteria on each square inch of the human body. If your skin has a total area of 1.7m squared, how many bacteria are you carrying around on your skin? 2.A flea jumps into the air and lands about 8.0 inches away, having risen to an altitude of 130 times its own height (comparale to about 650ft up for you). Assume a 45 degree launch. Compute the flea’s take-off speed.
- Mike Greco (age 20)
Elmhurst College, Villa Park, IL, USA
A:
Mike -

Well, I'm not going to do your homework for you, but I can give you a bit of help:

Number 1) What you know: (1)the number of bacteria per square inch and (2)the number of square meters total. So what's left? You need to know the number of square inches in a square meter. Well, there's 39.4 inches in 1 meter. Now we can just square these to give us 39.4^2 = 1550 square inches in 1^2 = 1 square meter. The rest is up to you.

Number 2) This is a case where you may have more information than you can actually use. Knowing that the flea goes up to 130 times its own height is really useless if you don't know how tall the flea is... you can't get an exact height from this. But you don't really need this info to solve the problem. Try breaking it down into two directions: up-down (or y) and side-to-side (or x) and you can get some useful equations:

(Remember that for any motion, you can say that
distance=velocity*time+0.5*acceleration*time^2, or d=vt+0.5at^2)

Since we know that the launch is at 45 degrees, the velocity in the side-to-side (x) direction is the same as in the up-down (y) direction. Try drawing a picture of this. You will see that the two directions make a triangle with the flea's take-off direction, where the x side and the y side are the same length (the velocity in each direction is the same). I'm going to call the flea's total velocity V and its x velocity (same as its y velocity) v.

In the x direction, there is no acceleration. So the equation is simply x=vt. In the y direction, the acceleration is due to gravity pulling down, with a=-g. This gives the equation y=vt-0.5*g*t^2. Now let's look at the point where the flea lands. At this point, the flea is at exactly the same height that he started at, so y=0. But he's gone forward by 8 inches (= d cm). Now you have two equations and two unknowns:

d=vt and 0=vt-0.5*gt^2

From here you should be able to solve for v on your own. As for V, you'll need to look back at your triangle. The two short sides are v, and the long side is V. So how do you from v to V? You can use the Pythagorean theorem: a^2+b^2=c^2, or v^2+v^2=V^2. Hopefully you can take it from here.

-Tamara

(published on 10/22/2007)