A Problem From Afar

My Dear Dear students from last year! It's been a while and I needed to know that you haven't stopped thinking and started memorizing random facts. I miss you all terribly and to show how much I miss you I shall pose a small question to you that should surely look like a "mole hill" to you when compared to your prior achievements. 

Is there a geometric proof of the algebraic statement below?

a² – b² = (a+b)(a-b)

If so, share your suggestion or proof by commenting to this post and remember, you'll have to communicate a visual idea through words because it'll be hard to share images when commenting. Goodluck!

Any Questions?

You can use this post for questions you may have regarding the topics we have covered in both geometry and trigonometry.

Homework due 05/20/09

So, given the radius of a circle:

1. Can you find the area of a sector if you know the central angle that the minor arc makes?

2. Can you find the area of a sector if you know the arc length that the sector intercepts?

(Note: The sector of a circle is like a pizza slice, in case you were wondering what a sector is!)

Homework due 05/15/09

Find the area of the triangle below. (Hint: You will need to find at least one angle, and I would suggest you employ the use of an altitude line.)

Homework due 05/12/09

You have the worksheet!

Homework due 05/11/09

1. On February 10, 1990, high tide in Boston was at midnight. The water level at high tide was 9.9 feet; later, at low tide, it was 0.1 feet. Assuming the next high tide is at exactly 12 noon and that the height of the water is given by a sine or cosine curve, find a formula for the water level in Boston as a function of time, measured in hours since midnight.

2. Of course, there's something wrong with the assumption in the problem above that the next high tide is exactly at noon. If so, the high tide would always be at noon or midnight, instead of progressing slowly through the day, as in fact it does. The interval between successive high tides actually averages about 12 hours 24 minutes. Using this, give a more accurate formula for the height of the water as a function of time.

Simple Harmonic Motion

So you obviously know that you have to plot the tangent functions graph using your unit circles and a table of values. Once you are done with that, you need to establish improvements/refinements to your method of getting precise measurements to craft a function that does a good job of modeling the "simple harmonic motion" of a mass suspended at the end of a spring. Think about:

(a) Number of oscillations in a given time period.

(b) The period (time taken for one oscillation).

(c) The lowest height, the greatest height and the equilibrium position.

(d) How you can utilize all the members of your group in an efficient way.

There may be other considerations and you should certainly discuss them using the blog. On Friday, without a single minute to waste, you must be ready to commence with your function modeling exercise. The end goal is that you come up with a function that fairly accurately models the motion of the suspended mass.