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by **Fredfredburger** » Sat Apr 28, 2012 9:05 pm

I need a little help with a couple of these.

Regional #36
In triangle ABC, m∠ BAC=70° and m∠ ABC=50°. Point H lies on segment AB such that segment CH is an angle bisector of ∠ ACB. Point M lies on segment BC such that BM=BH. Find m∠ CHM

Ans: 35°

Regional #55
The complex numbers w=2+5i and z=3-4i are graphed as vectors. What is the measure of the angle formed between them?

Ans: 121.3°

State #18
Triangle ABC exists such that AB=AC. P lies on AC and Q lies on BC such that AP=AQ. Find m∠ PQC if m ∠ BAQ=30°.

Ans: 15°

State#22
Mr. White and his dog, Lady, try to speed walk 2 miles each day when the weather permits. The following table shows Mr. White's speed for each half mile walked. Find Mr. White's average walking speed.

Distance Walking speed
First 1/2 mile 4.3 mph
Second 1/2 mile 3.0 mph
Third 1/2 mile 3.3 mph
Fourth 1/2 mile 2.7 mph

Ans: 3.30 mph
(I took the geometric mean and got it wrong so I tried the harmonic mean and it was also wrong. The only way I've been able to produce the answer is when I take the arithmetic mean of the first and second and then the third and fourth and then take the geometric mean of those two. It might be the quadratic mean, but I forgot that one. Can someone please tell me how you know when to apply which mean with problems like this one and the trapezoid ones.)

State #29
Roland Cubes tossed a fair die three times. What is the probability that the face with six dots comes up at least two times?

Ans: 7.4%
(I forgot the general formula)

State #54
The equation sec^2(x) + cos^2(x)=2 can be simplified to the equation:

Ans: cos^2(x)=1

State#59
How many three digit numbers that are not divisible by five can be formed from the digits 0,1,2,3, and 5 without repetition.

Ans: 27

by **007math** » Sat Apr 28, 2012 10:22 pm

Fredfredburger wrote:Regional #55
The complex numbers w=2+5i and z=3-4i are graphed as vectors. What is the measure of the angle formed between them? Ans: 121.3°

We graph things in the Cartesian plane in which there is a x-axis and a y-axis. In the complex plane, the current x-axis becomes the real axis (having real numbers) and the current y-axis becomes the imaginary axis. Therefore, $2+5i \implies <2,5>$ and $3-4i \implies <3,-4>$ . To find the angle between these vectors, imagine the two right triangles formed. The angle is therefore $\arctan{\frac52} + \arctan{\frac43} = 121.3$

by **redhawk2015** » Sun Apr 29, 2012 6:29 pm

Fredfredburger wrote:State #29
Roland Cubes tossed a fair die three times. What is the probability that the face with six dots comes up at least two times?

Ans: 7.4%

Conditional Probablility.

Case 1: A 6 comes up three times.
The probability of this is $(1/6)*(1/6)*(1/6) = 1/216 \thickapprox 0.46\%$

Case 2: A 6 comes up two times.
The probability of this is (1/6)*(1/6)*(5/6) = 5/216

. You have to multiply this by three because there are three different ways in which two separate rolls can come up as 6 (rolls 1 and 2, rolls 1 and 3, or rolls 2 and 3). This gives $15/216 \thickapprox 6.94\%$ .

Add the two together for a total of $0.46\% + 6.94\% = 7.4\%$

by **ohernandez** » Sun Apr 29, 2012 8:10 pm

Fredfredburger wrote:State#22
Mr. White and his dog, Lady, try to speed walk 2 miles each day when the weather permits. The following table shows Mr. White's speed for each half mile walked. Find Mr. White's average walking speed.

Distance Walking speed
First 1/2 mile 4.3 mph
Second 1/2 mile 3.0 mph
Third 1/2 mile 3.3 mph
Fourth 1/2 mile 2.7 mph

Ans: 3.30 mph

The harmonic mean is the best way to average rates of things - if you have quantities in the form "unit1's per unit2" (miles per hour, price per earnings, etc.)
$\displaystyle\frac{4}{\frac{1}{4.3}+\frac{1}{3.0}+\frac{1}{3.3}+\frac{1}{2.7}}\approx 3.23 mph$
The real answer is E. 3.23 mph

by **Salvadoramerican** » Mon Apr 30, 2012 7:02 am

the first one is a bit difficult to explain without a drawing but i'll trie first find the missing angle which is 60 the bisector makes it 30. Remember that triangle HMB is isoceles then the two angles are 65 because 180-50=130/2=65. and with the 30 degrees we know angle AHC=80 so then angle CHM=180-65-80=180-145=35

by **Salvadoramerican** » Mon Apr 30, 2012 1:12 pm

$\sec ^2 x+\cos ^2 x=2 \Rightarrow \frac{1}{\cos ^2 x} + \frac {\cos ^4 x}{\cos ^2 x}=2 \Rightarrow \frac{1+\cos ^4}{\cos ^2}=2$ $\Rightarrow 1+\cos ^4 x=2\cos ^2 x$ $\Rightarrow \cos ^4 x-2\cos^2 x+1=0 \Rightarrow \left ( \cos ^2 x-1 \right ) ^2=0$ $\Rightarrow \sqrt{Ans}=\sqrt{0} \Rightarrow \cos ^2 x =1$
double check

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