We already saw the busiest shop in the world (# of customers approach infinity when it gets real cold).

How strong is gravity in Albany?

#7 The height of a swimmer’s dive off a 10-foot platform into a diving pool is modeled by the equation $y = 2x^2 - 12x + 10$, where x represents the number of seconds since the swimmer left the diving board and y represents the number of feet above or below the water’s surface. What is the farthest depth below the water’s surface that the swimmer will reach?

Physics or sports folks out there – how horrible is that model? I assume that when the diver hits the water, the acceleration changes dramatically, right? And for the part before the water an acceleration of 4ft/s per s is more like moon gravity than Albany gravity (even though I hear there is strange stuff up there). Can anyone supply more detail?

Also note the subtle error in the description of y.

(for kiddies: answer is the vertex of the parabola – (3,-8) so 8 feet below the water’s surface.)

#30 [ corrected the wrong #: #8] Farmington, New York, has plans for a new triangular battlefield park. If plotted on a coordinate grid, the vertices would be A(3,3), B(5,-2), and C(-3,-1). However an evil wizard a tract of land has has seized control become available that would enable and the wizard the planners to plans to turn it into a strip mall increase the size of the park, which is based on the following transformation of the original triangular battlefield park, $R_{270} \circ D_2$

• my version would at least make it amusing. Theirs is stupid.
• they treat their notation as standard (not requiring explanation). I have seen enough transformational geometry texts to know that it is not. having New York State official (but not accepted by the wide world) mathematical nomenclature is unacceptable.
• they don’t specify the center of dilation. they don’t specify the center of rotation.
• this question fails to assess something that matters: whether kids understand how to compose. If they do the rotation first, no problem… (although, in defense of Albany, there is a multiple choice item, #9, that did exactly that, and with the trap answer.)

kiddies, they meant the center of each transformation to be at the origin, and we work from do the first to the second, iow, the dilation comes first:

$(x,y) \mapsto (2x,2y) \mapsto (2y,-2x)$ and the coordinates become (6,6), (-4,-10), (-2,6).

#10 A central angle of a circular garden measures 2.5 radians and intercepts an arc of 20 feet. What is the radius of the garden?

Stop right there! We don’t measure angles on the ground in radians. When we do use radians, they appear as multiples of π!

The answer is 50 feet. The author is an idiot.

1. January 28, 2009 pm31 8:01 pm 8:01 pm

I don’t think D_2 for a dilation is just nonstandard, it’s *wrong*.

It’s already in use for dihedral groups!

2. January 29, 2009 am31 7:17 am 7:17 am

Physics model – pretty horrible.
Store sales model – pretty horrible.

It was probably created by someone with an imperfect grasp of math, possibly a grad student or a new teacher who hasn’t really got the basics nailed down but feels he needs “real-world” problems.

As you point out, problems like this can only be dealt with if you have an equally imperfect understanding of the world. Prior knowledge messes up everything.

January 30, 2009 am31 6:45 am 6:45 am

“Also note the subtle error in the description of y.” The only thing I notice is that y should be the location of the sole of the diver’s foot at time x, with respect to water’s surface (since its a 10 ft board, so at x=0 it is the sole on the board, whereas for diving depth you probably wouldn’t measure to the sole of the foot (the highest point, once the diver is in … My guess is you had something else in mind. What?

4. January 30, 2009 pm31 7:40 pm 7:40 pm

Um, I have less a problem with gravity being weak that with it pointing up! This diver will be in one heck of a pickle when he or she hits the surface again at x=5.

5. January 31, 2009 am31 7:15 am 7:15 am

I wouldn’t have trouble with a piecewise function that has endpoints at the water’s surface, positive acceleration, etc. I wish they had left it at that or even taken the diver out of it all together.

But that wouldn’t have been all real-world enough, would it?