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
(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
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.
(republished on 07/18/06)