Sorry about not having the leaderboard completely up yet, the new AC charger to my laptop is in the mail as we speak...
Just so yall know, I got a 178 on this one. Did 41 and got 3 wrong. I am sooo slow on these tests...
I decided to go to the end afterward just to see how good I was and now I come bearing questions...
#50 est prob: 21 X 23 X 25 = ______________
(I tried cubing 23, but I had no clue how to do this quickly...)
#66 4^11 / 6 has a remainder of _____________
(I knew that the remainder had to be 0 2 or 4 but I couldnt find a pattern here to solve it)
#71 the horizontal asymptote of f(x) = x-3/x^2+x-12 = ____________________
(um... I tried factoring and I got 1/x+4, in which case I thought x != -4, but wrong. I tried arbitrarily to find the derivative as well which made 1/2x+1... another dead end.)
#72 f(x) = sin(x), f'(pi/4) =
(I hate trig.... I think the derivative of the sine is the cosine, and that pi is about 180 degrees so that means that 1/4 of pi radians is 45 degrees. That makes the answer make more sense now that I look at it, but how do we know whether it is negative or not? Or for that matter what figure we are supposed to define in the problem. I think sqrt(2)/2 defines the x and y coordinates of a 45/45/90 triangle in the first or third quadrants... Somebody explain the logical process and answer here. PLEASE!!!)
#74 Change 11/125 into a base 5 decimal.
(I assumed the method of rewriting the fraction in base 5 beforehand and missed the problem. I said hmmmmm..... 11/125 = 21/100 base 5 and stupidly wrote .21, when the answer should've been .021. I am guessing that the digits here are represented as 5^-1, 5^-2, and 5^-3. What is the shortcut here?)
#79 The odds of losing are 5/11, the probability of winning is: _____________
(I am utterly stumped here. Where did 11/16 come from?)
No sweat about the leaderboard. As long as the test is there, I'm happy, lol.
Ok, you're questions.
50. 21 * 23 * 25
I did 23 cubed. On the math-magic.com website there's a page about which cubes and higher powers you should memorize, and cubes up to 23 were on the list. If you don't have them memorized, do this: 21 * 23 is really close (only one off) to 22^2, which is 484, 484 * 25 is now easy because it's divisible by four. You'd get 12100 which is only 25 off!; by the way, 23 cubed is 12167.
66. 4^11 / 6 has a remainder of ...
ok, this one was a little tricky for me too. I kept looking for a power of 4 that would give a remainder of one or 2 when divided by 6, but what i noticed was, 4^1 / 6 has a remainder of 4, 4^2 / 6 has a remainder of 4, 4^3 / 6 has a remainder of 4; So, i just decided that I'd put 4 and leave it at that, which was lucky because it was the right answer.
71. Horizontal Asymptotes are a bit different from vertical asymptotes. With horizontal asymptotes, you're only concerned about the behavior of the function as x approaches really large values. So, you only need to worry about the largest degree of the polynomials. Because, say you had (x-3)/(x-5), as x approaches values of, say, 100,000 and above, the three and 5 really aren't going to make that much difference, and you'll approach something closer to one. ok, that was just me rambling, so if you don't understand any of that, it's ok.
Here's what you have to know about horizontal asymptotes:
Case 1) If the degree of the numerator is less than the degree of the denominator, the line y = 0 will be the horizontal asymptote.
Case 2) If the degree of the numerator equals the degree of the denominator, then the line y = a/b is a horizontal asymptote, where a and b are the leading coefficients of the numerator and denominator respectively.
Case 3) If the degree of the numerator is greater than the degree of the denominator, there's no horiz. asymptote.
Case 1 is used in this case.
I'll finish up later, if nobody else does. Gotta go to tennis camp.
I don't know how much time I have but I will try to answer all of your questions. I will pick up where Vinay left off. But I want to revisit #66.
66) 4^11 divided by 6...This is how I would do this problem. I split it up like so: (4^3)^3 x 4^2. Now, take the remainder of each term gives (64 MOD 6)^3 x (16 MOD 6) = 4^3 x 4. Where are still looking for the remainder so we can take this whole term (4^3 x 4) MOD 6 x 4 = 64 MOD 6 x 4 = 4 x 4 = 16. So, 16 MOD 6 = 4. It's a lot of work writing all of this down, but it only takes a couple of seconds to do in your head. Now, there is a different short cut for the problem like this on Test #3. But I want you to take the test first.
72) You are right on the money on #72 Zack. The answer is sqrt(2)/2. PI/4 = 45 degrees. So we have a 45-45-90 triangle. That gives us sides of 1 and 1, with a hypotenuse of sqrt(2). The derivative of sin(x) is cos(x), so we are looking for cos(45), which is adjacent/hyptenuse or 1/sqrt(2). You can't have square roots in the denominator, so we have to simplify to sqrt(2)/2. You know it is positive because 45 degrees (or PI/4) is in the first quadrant. If I would have said 3Pi/4, then the answer would have been negative because it would have been in the second quadrant).
74) Very close Zack. 11/125 into base 5 decimal. 11 base 10 = 21 base 5. To find out how many decimal places you use, you have to look at the denominator. 125 = 5^3. The exponent tells you have many decimal places. So we have 3. The answer is taken to 3 decimal places, or .021.
79) You need to look under my section on Probability and Statistics. Basically, there is a difference between odds and probability. Probability = Number of desired outcomes/Number of total possibilities. Odds = number of desired outcomes/Number of undesireable outcomes. If I said I had a 1:1 chance of winning, we are talking about odds. If I said I had a 50% (or 1/2) chance of winning, I am talking about probability. Notice, both say the same thing. So in our example we have the odds of losing being 5/11. That means 5 = desired outcomes and 11 = undesired outcomes. This also means we have 5+11 total outcomes or 16.
So we need to change 5/11 to probability, making it 5/16. This is the probability of losing. To find the probability of winning, we subtract this number from 1. 1 - 5/16 = 11/16. This is the answer. If you have any other questions, just ask.
The test and answer key is posted on the website. Lucky for me that I store most of my math stuff in the free memory that gmail and hotmail provide.
Thanks to people like you awesome people at these math like forums, I stand a chance next year att regionals, despite the fact that the entire Medinna Valley team will be back next year in full force. I cant wait to give them a run for their money!
Thanks again for the help, all those problems make sense now.
