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by **strr** » Sat Mar 31, 2012 6:42 pm

I'm sure most of these problems are simple, but I'm learning :)

7. Noah Sense has 42 coins consisting of pennies, nickels, dimes, and quarters. He has twice as many nickels as pennies, three less dimes than nickels, and three more quarters than pennies. How much money does he have? Ans = $4.37
I would like to know how to set this up, and how to go about setting up similar problems like this. I know it is a simple problem, but for some reason I have the hardest time understanding how to work them out.

13. Find the angle of rotation, theta (I don't know how to get the symbol, sorry) (nearest tenth degree), where theta is between zero and 90 degrees, such that the conic 2x^2 + 12xy + 18y^2 - 3y = 5 contains no xy term in its equation. Ans = 71.6 deg

22. Let p and q be the real roots of x^2 - 2x - 8 = 0, where p > q. Find p^3 + 2p^2q^2 + pq^3. Ans = -32
I know how to work that out by factoring and plugging in, but there must be some shortcut/alternate way to decrease the time to solve this.

23. Bea Debest, Ima Slo, and Betsy Luzes run in a 200 meter race. When Bea crosses the finish line, Ima is 10 meters behind Bea. When Ima crosses the finish line, Betsy is 10 meters behind Ima. If all 3 runners ran at a constant speed, how far was Betsy from the finish line when Bea won the race? Ans = 19.5 meters

35. Let A = [1, -2, 0, x] and A^-1 = [1, 4, y, 2]. Find x + y. These are supposed to be matrices, sorry! Ans = 1/2

36. How many asymptotes exist of h(x) = (x+10)/abs(x)? Ans = 3. I know how to graph this one and count, but is there a way to look at the function and see how many asymptotes there will/might be?

40. The Brite Lite Company produced 5000 100-watt bulbs of which 50 were defective. The Brite Bulb Company produced 3000 100-watt bulbs of which 100 were defective. A bulb was chosen at random from the 8000 bulbs and turns out to be defective. What is the probability that the bulb came from the Brite Lite company? Ans = 1/3 %

41. A pair of dice are rolled. What are the odds that the roll comes up as a 2, 5, 6, 10, or 12? Ans = 7 to 11

42. I'm not sure how to type that one out on here, but if anyone has access to this test and knows how to do this question, that would be amazing!

55. The polar graph of r = 2sin(3theta) is symmetric to the: Ans = line theta = pi/2

Any help is greatly appreciated, thank you!

by **88bobcat** » Sat Mar 31, 2012 11:44 pm

strr wrote:42. I'm not sure how to type that one out on here, but if anyone has access to this test and knows how to do this question, that would be amazing!

This one has been covered a couple of times already. Refer to this URL: http://www.texasmath.org/forum/viewtopic.php?f=4&t=3143.

strr wrote:7. Noah Sense has 42 coins consisting of pennies, nickels, dimes, and quarters. He has twice as many nickels as pennies, three less dimes than nickels, and three more quarters than pennies. How much money does he have? Ans = $4.37
I would like to know how to set this up, and how to go about setting up similar problems like this. I know it is a simple problem, but for some reason I have the hardest time understanding how to work them out.

Substitute and solve to get

P=7, N=14, D=11, Q=10

or

$Q=\$2.50$

$D=\$1.10$

$N=\$0.70$

$P=\$0.07$

Total is $\boxed{\$4.37}$

by **88bobcat** » Sat Mar 31, 2012 11:54 pm

strr wrote:13. Find the angle of rotation, theta (I don't know how to get the symbol, sorry) (nearest tenth degree), where theta is between zero and 90 degrees, such that the conic 2x^2 + 12xy + 18y^2 - 3y = 5 contains no xy term in its equation. Ans = 71.6 deg

Larry White has stated that conics and their rotations will no longer be included in the tests.

However, you can find plenty of information on this with a simply on-line search. Have a look at http://www.sparknotes.com/math/precalc/conicsections/section5.rhtml.

strr wrote:22. Let p and q be the real roots of x^2 - 2x - 8 = 0, where p > q. Find p^3 + 2p^2q^2 + pq^3. Ans = -32
I know how to work that out by factoring and plugging in, but there must be some shortcut/alternate way to decrease the time to solve this.

You should first note that

is a common factor in each of the terms of p^3q+2p^2q^2+pq^3

.

Factoring yields (pq)(p^2+2pq+q^2) = (pq)(p+q)^2

.

Now you can simply apply Vieté to see that pq = -8

and

.

Hence, $p^3q+2p^2q^2+pq^3 = (-8)(2)^2 = \boxed{-32}$

by **88bobcat** » Sun Apr 01, 2012 12:04 am

strr wrote:36. How many asymptotes exist of h(x) = (x+10)/abs(x)? Ans = 3. I know how to graph this one and count, but is there a way to look at the function and see how many asymptotes there will/might be?

To find asymptotes, you must find out what happens to the function when its independent variable approaches infinity in either direction and you must find out the conditions of the independent variable that cause the function to approach infinity.

Because of the absolute value function, you must consider the limits of this separately:

$\displaystyle \lim_{x_+ \rightarrow \infty} \frac{x+1}{|x|} = \lim_{x_+ \rightarrow \infty} 1 + \frac{1}{x} = 1$

$\displaystyle \lim_{x_- \rightarrow -\infty} \frac{x+1}{|x|} = \lim_{x_- \rightarrow -\infty} -1 + \frac{1}{x} = -1$

$\displaystyle \lim_{x_{\pm} \rightarrow 0} \frac{x+1}{|x|} = \infty$

by **88bobcat** » Sun Apr 01, 2012 12:14 am

strr wrote:41. A pair of dice are rolled. What are the odds that the roll comes up as a 2, 5, 6, 10, or 12? Ans = 7 to 11

With a pair of dice, the probability of rolling $n \in \{2,3,4,5,6,7\}$ is $\displaystyle \frac{n-1}{36}$ .

Similarly, the probability of rolling $n \in \{8,9,10,11,12\}$ is $\displaystyle \frac{13-n}{36}$ .

$\displaystyle P(2) = \frac{2-1}{36}$

$\displaystyle P(5) = \frac{5-1}{36}$

$\displaystyle P(6) = \frac{6-1}{36}$

$\displaystyle P(10) = \frac{13-10}{36}$

$\displaystyle P(12) = \frac{13-12}{36}$

Now, the probability is $\displaystyle \frac{1}{36}+\frac{1}{9}+\frac{5}{36}+\frac{1}{12}+\frac{1}{36} = \frac{7}{18}$

Hence, the odds are $\displaystyle \boxed{\frac{7}{11}}$ .

by **strr** » Sun Apr 01, 2012 12:10 pm

Thank you so much for your help.

by **88bobcat** » Sun Apr 01, 2012 1:57 pm

strr wrote:35. Let A = [1, -2, 0, x] and A^-1 = [1, 4, y, 2]. Find x + y. These are supposed to be matrices, sorry! Ans = 1/2

$\displaystyle \bar{A}^{-1} = \frac{\bar{I}}{\bar{A}}$

$\displaystyle \bar{I} = \bar{A} \cdot \bar{A}^{-1} = \begin{bmatrix} 1 & -2 \\ 0 & x \end{bmatrix} \cdot \begin{bmatrix} 1 & 4 \\ y & 2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

1 - 2y = 1

and

So, $\displaystyle x = \frac{1}{2}$ and y = 0

$\displaystyle \longrightarrow x+y = \boxed{\frac{1}{2}}$

by **sxk1693** » Tue Apr 03, 2012 4:58 pm

For 22 i found the roots and plugged it into the equations.
For 55 i graphed it.

by **strr** » Fri Apr 06, 2012 10:53 am

Just checking if I could get an answer to this:
23. Bea Debest, Ima Slo, and Betsy Luzes run in a 200 meter race. When Bea crosses the finish line, Ima is 10 meters behind Bea. When Ima crosses the finish line, Betsy is 10 meters behind Ima. If all 3 runners ran at a constant speed, how far was Betsy from the finish line when Bea won the race? Ans = 19.5 meters

by **PokémonMaster** » Fri Apr 06, 2012 12:09 pm

For the race question,2nd place was 95% as fast as 1st place, and 3rd place was 95% as fast as 2nd place.Using this,you can square 95%(how fast each person is in relation to the racer directly in front of them) to find that when 1st place finished, 3rd place was 90.25 percent done.Multiply 90.25* 200(the length of the race), and you get 180.5, the distance 3rd place would have traveled. Subtract this from 200 to get 19.5, which is the answet

by **strr** » Fri Apr 06, 2012 12:23 pm

Oh okay. Thanks!

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