Monday, March 10, 2014

Day 52: Right six, CARE, tightens, gnarl. Jiggy Jiggy, Fifty, crest. Right three minus

Go full size:

The 7.6-mile-long Mt. Washington Road is lined with trees, drop-offs, and winds on the way to the 6,288-foot summit. Travis Pastrana in a 2011 Subaru Impreza WRX STI rally car, with an officially timed run of 6 minutes 20.47 seconds.

There are questions all over the place.
1. Average speed
2. Average slope of the road.
3. What do the calls mean?
This one includes rally notes system details. "Increases the radius, decreases the radius" "Can you take it wide open?" "If the notes are wrong, we go off a cliff."

Sunday, March 9, 2014

Day 51: Pythagorean Triples

If you've worked with the Pythagorean Theorem, you've come across some integer solutions.

3-4-5 and 5-12-13 come to mind, especially if you've studied for the SAT; 8-15-17 is another good one.
How many can you find?
Is there a way to find as many as you want?

(For those who haven't seen this trivia snippet)

pick distinct positive integer values for u and v: perhaps 3 and 4.
The three sides are 2*u*v, u² + v², and |u² - v²|, thus 24, 25, 7
The hypotenuse is simply the longest of the three sides, but won't be generated by the same expression all the time.

Saturday, March 8, 2014

Day 50: Dividing by a fraction

Why is dividing by a fraction equivalent to multiplying by the reciprocal ... Here's one explanation, from my other blog: We "invert and multiply", "multiply by the reciprocal" or insist on using the fraction key because we can't remember or were never really taught the reasons or the algorithm. Is there a simple explanation for the method we old farts memorized years ago in third or fourth grade? Why does it work?
Let's start with a problem: $\frac{3}{4} \div \frac{5}{6}$ and change to a compound fraction: $\dfrac{\frac{3}{4}}{\frac{5}{6}}$

Now what? Dividing by a fraction is confusing, but dividing by 1 is obvious. So we turn $\frac{5}{6}$ into unity by multiplying by its reciprocal. Of course, you can't just multiply part of our problem by $\frac{6}{5}$ without changing its value, so we multiply by 1: $\dfrac{\frac{6}{5}}{\frac{6}{5}}$

All in one image: $\dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \rightarrow \dfrac{\dfrac{3}{4}}{\dfrac{5}{6}} \cdot \dfrac{\dfrac{6}{5}}{\dfrac{6}{5}} \rightarrow \dfrac{\dfrac{3}{4} \cdot \dfrac{6}{5}}{\dfrac{1}{1}} \rightarrow \dfrac{3}{4} \cdot \dfrac{6}{5} \rightarrow \dfrac{18}{20} \rightarrow \dfrac{9}{10}$

Divide by one. Seems simple to me.

Friday, March 7, 2014

Day 49: Rectangle made up of Squares

"A perfect fit with no overlapping"

Take the squares as defined below and fit them into one big rectangle with no gaps or spaces between the squares or in the corners.

The squares have sides of the following lengths 2, 5, 7, 9, 16, 25, 28, 33, & 36.

Thursday, March 6, 2014

Day 48: Pizza, pizza

Pizza ... makes you hungry doesn't it?

What if you had ten toppings available. How many different two-topping pizza variations can you make ...

source:

Wednesday, March 5, 2014

Day 47: The One Where We Label Axes.

Why are coordinate axes perpendicular?
Do they have to be?
What if there are more than two dimensions?