<p>1) If 0 <( less than and equal to ) x < ( less than or equal to) y and (x+y)^2 -(x-y)^2 >(or equal to) 25. What is the least possible value of y?</p>
<p>also :</p>
<p>2) photo</a> 5 | Flickr - Photo Sharing!</p>
<p>3)photo</a> 4 | Flickr - Photo Sharing!</p>
<p>4)photo</a> 3 | Flickr - Photo Sharing!</p>
<p>5)photo</a> 2 | Flickr - Photo Sharing!</p>
<p>6)photo</a> 1 | Flickr - Photo Sharing!</p>
<p>Thank you in advance!</p>
<p><a href=“http://talk.collegeconfidential.com/sat-preparation/159002-hard-math-problem.html[/url]”>http://talk.collegeconfidential.com/sat-preparation/159002-hard-math-problem.html</a></p>
<p>Number 6:</p>
<p>Solution by plugging in numbers: Let’s plug in some simple values for x.</p>
<p>x = 0: 8k = m
x = 8: 0 = 64 – 40k + m</p>
<p>Substituting 8k for m in the second equation yields 0 = 64 – 32k, so that 32k = 64, and k = 2. Finally, m = 8k = (8)(2) = 16. </p>
<p>Solution by equating like terms: Multiply out the left hand side (FOIL) to get</p>
<p>x^2 – kx – 8x + 8k = x^2 – (k + 8)x + 8k</p>
<p>Setting the coefficient of x on the left equal to the coefficient of x on the right yields -(k + 8) = -5k, or k + 8 = 5k, or 4k = 8. So k = 2. Equating the constant terms on left and right yields 8k = m. Substituting 2 in for k gives m = (8)(2) = 16, choice (B).</p>
<ul>
<li>Quickest solution: The left hand side is 0 when x = 8 and x = k. The coefficient of x is the negative of the sum of these roots, so 5k = k + 8, or 4k = 8. So k = 2. The constant term is the product of these roots, so that m = 8k = (8)(2) = 16, choice (B).</li>
</ul>
<p>Note: For the previous solution we used the following general theory:</p>
<p>Let r and s be the roots of the quadratic equation x^2+bx+c=0. Then</p>
<p>b=-(r+s) and c=rs.</p>
<p>Number 5 was done here: <a href=“http://talk.collegeconfidential.com/sat-preparation/1488170-jan-2013-math-question.html[/url]”>http://talk.collegeconfidential.com/sat-preparation/1488170-jan-2013-math-question.html</a></p>
<p>Number 4:</p>
<p>The radius of each circle is 24/4 = 6. So the circumfeence of each circle is 12pi. </p>
<p>Now, note that angle X is a 120 degree angle. To see this draw line segment XY and the perpendicular bisector of this segment between the points of intersection of the two circles. Each of these triangles has a leg of length 3 and hypotenuse 6. Since the hypotenuse is double this leg, the angle between these two sides is 60 degees.</p>
<p>So the darkened arc of each circle contains 360 - 120 = 240 degrees. This is 2/3 of 360 degrees. So the darkened arc of each circle is (2/3)(12pi) = 8pi</p>
<p>Now we double this since there are 2 such circles to get 16pi, choice (B).</p>
<p>Number 3:</p>
<p>Triangle FGA is isosceles so the base angles are congruent. Thus each base angle is (180 - 22)/2 = 158/2 = 79. </p>
<p>So x=79 as well, choice (A).</p>
<p>Note: BC and FE are parallel lines cut by the transversal BF. Therefore the measure of angle FBC is equal to the measure of angle AFE.</p>
<p>Number 2:</p>
<p>For every 4 inches along the paper strip, we need to add 5 inches. So we can set up the following ratio.</p>
<p>length along strip<strong>4</strong>80
length in bold__<em>__5</em>_x</p>
<p>We now find x by cross multiplying and dividing.</p>
<p>4/5 = 80/x
4x = 400
x = 400/4 = 100.</p>
<p>I have not read all the questions (since Dr Steve answered them) but I think that for number Two, the explanation might be a bit off. </p>
<p>
</p>
<p>I believe it should be for every 4 inches along the paper strip, we need to add 1 inch (because the we add two sides of the triangle and take one away) to the original number. </p>
<p>Or we could say … every 4 inches become 5 inches with the addition. Thus 80 becomes 100. </p>
<p>If I am wrong, chalk it to look at it very quickly. Either way, the answer is indeed 100. That I know :)</p>
<p>Oh, yes, to anyone who looks at past ETS/TCB questions, START by copying part of the question and using google search on this forum. Most questions have been answered numerous times and in different ways. </p>
<p>It really beats reposting an image of the question, and saves you time. </p>
<p>PS Be careful with answers posted in sites like yahoo. The answers are often provided by people who know even less than you do. :)</p>
<p>@xiggi</p>
<p>I didn’t mean to imply that we should add 5 inches to 4 inches. But now that I look at it I see how that sentence could be taken that way. </p>
<p>I actually don’t think I’m completely happy with any of the ways either of us have phrased this (although I think they are all correct with my original phrasing perhaps the most ambiguous). </p>
<p>Maybe the best way to say it is that we’re replacing each 4 inches by 5 inches.</p>
<p>In any case, I think the ratio makes it pretty clear what we mean.</p>
<p>Writing clear explanations are often harder than solving the problems. :)</p>