# A Rate of change and a differentials problem

1. Nov 19, 2007

### aaroffl

First off i'd like to say Hi to the forums, hehe. I didn't really see a new member area but I suppose this will do. Right now I'm kind of struggling with these two problems that I recently took a quiz on and didn't do so well. I've been trying to figure out how to work them out but i just dont really get it. if anyone could help out it'd be awesome.

(1)

The position (in cm) of a particle along the x-axis at time t(in s) is given by X(t) = t^3-6t^2+8t, t is greater than or equal to 0.

the question:
When is the particle moving to the left? the right?
Acceleration of the particle when the speed is zero.
Total distance traveled from t=0 to t=2.

(2)

( and the one that REALLY gets me)
A person standing 30ft from the base of a building measures the angle of elevation to the top of the building as being 75 degrees with an error of +or- 1.2 degrees.

What is the max error in estimating the height of the building, and the percentage error?

This one has taken me quite a while, I am thinking it has something to do with the law of sines, but when i try it with that method i'm getting a completely unreasonable answer. I kind of have the right idea, i hope, but for some reason i'm just not seeing it.

Thank you in advance for those who wish to take on these.

2. Nov 19, 2007

### ozymandias

(1.) Let x' be the first derivative with respect to t, x'' the second.
a.) Right when x'>0, left when x'<0.
b.) Find at what time t the velocity x'=0, and then substitute this into x''.
c.) compute x(2)-x(0)

(2.) If you know the angle of elevation, a, the height of the building is given by
h = 30ft * tg(a)
Now the error in your case is given by Delta a = +-1.2 degrees = +- 0.02 radians, and the resulting error in h is:
$$\Delta h = (1+\tan(a)^2) \Delta a$$
(If you need further explanations on that one, ask.)
Now you merely need to substitute.
Good luck!

--------
Assaf
Physically Incorrect