Math question

Test Preparation SAT Preparation
1

<li>The sum of the positive odd integers less than 100 is subtracted from the sum of the positive even integers less than or equal to 100. What is the resulting difference?</li>
</ol>

<p>Answer: 50</p>

<p>How do you get 50 without adding up all the integers?</p>

<li>A large solid cube is assembled by gluing together identical unpainted small cubic blocks. All six faces of the large cube are then painted red. If exactly 27 of the small cubic blocks that make up the large cube have no red paint on them, how many small cubic blocks make up the large cube?</li>
</ol>

<p>Answer: 125</p>

<li>In the xy-coordinate plane, the graph of y = -x^2 + 9 intersects the line L at (p, 5) and (t, -7). What is the least possible value of the slope of L?
(a) 6
(b) 2
(c) -2
(d) -6
(e) -10</li>
</ol>

<p>Answer: (d) -6</p>

2

<p>First off, 12love, that's 3 questions ;)</p>

<ol>
<li> You can do this with formulas. Don't. The best way to do this is to write out the sums (not all the terms, just see the pattern) and look at it:

2 + 4 + 6 + 8 + 10 + . . . 100

1 + 3 + 5 + 7 + 9 + . . . 99</li>
</ol>

<p>So reading down, each of the individual differences gives 1. How many are there?</p>

3

<p>LOL. yeah. There are three questions. I, at first, wrote one question, but I editted it and added two more.</p>

<p>Thanks. That was a good explanation.</p>

4

<ol>
<li> First, it's about cubes. So start by thinking about cubes made of cubes. You played with blocks a looong time ago, right?<br></li>
</ol>

<p>The number of blocks along each dimension must be the same (because the small blocks are cubes). So the side length (in terms of block sides) will be 1 (well, that's out), 2, 3, 4, 5, 6, . . . </p>

<p>Now the faces are painted. So to "escape" being painted, a cube needs to be on the inside of the whole block. The large solid cube must consist of a complete "coating" of blocks on the outside of the set of completely unpainted blocks. There are 27 unpainted blocks, so that's a cube of 3 x 3 x 3 blocks. To "coat" them, you'll need a cube that is 5 x 5 x 5. (Do you see why 4 x 4 x 4 doesn't work?). So that gives 125.</p>

<p>You will learn a lot that's useful for the SAT by thinking about other possible arrangements of the cubes, how many are painted and how many unpainted, how many have 2 sides painted, how many have 3 sides painted, could any have 4 sides painted, etc.</p>

5

<p>^^^beat me to it</p>

6

<ol>
<li> This one looks ugly at first, but it's actually not. The key to this one is to draw a graph, unless you are unusually good at visualizing things in your head. Start with the graph of -x^2 + 9. It's an inverted parabola, shifted up by 9.<br></li>
</ol>

<p>Now, the line L intersects the parabola at (p, 5) and (t, -7). The 5 and -7 haven't been picked at random. They're set up to make this problem solve neatly. (You don't need to know that in advance, but it might be a useful mindset to take into the SAT.)</p>

<p>Start with the intersection at (p, 5). This point must be on both the line and the inverted parabola. There are only 2 possible x values on the parabola where the y value is 5. These are 2 and -2. So the line L either goes through (2, 5) or (-2, 5). </p>

<p>Now, go to the intersection at (t, -7). Again, there are only 2 possible x values on the parabola that give this y value. These are 4 and -4. So the line L either goes through (4, -7) or (-4, -7).</p>

<p>Now, you need to find the least possible value of the slope. Keep in mind that in common parlance, the least slope would be the least steep. But that's not the mathematical meaning of the least slope! If the slope can be negative, then the steepest line with a negative slope has the least slope (because it's the smallest number, even though its absolute value isn't the smallest).</p>

<p>This is where your graph will come in handy. Looking at it, you can see that the steepest sloped line is the one running through (2, 5) and (4, -7). You could also do this from the points written out in a list, but I think that's harder. The only thing now left to do is to compute the slope of the line through (2, 5) and (4, -7).</p>

<p>Please let me know if you have questions about the last 2.</p>

7

<p>Ok. I got the last two! Thanks for your help!!</p>

8

<p>You're welcome. Good luck on the real thing!</p>

9

<p>another way for 3 is to solve for x using the equation.
you'll get (2,-2) and (4,-4) for p and t.
((+-)2,5)
((+-)4,-7) are the points</p>

<p>then use (y1-y2)/(x1-x2) for slope</p>

<p>numerator -7-5 = -12
you would need the denominator to be the smallest positive to get the smallest slope since the numerator is a negative 4-2 = 2</p>

<p>-12/2 = -6</p>

10

<p>wouldnt the intersection be at 2 and -7 and 4 and 5 ?!</p>

11

<p>its a parabola so for there are 2 x values for each y.</p>

12

<p>Quix, if you say a little more about your thinking, I might be able to clarify it for you.</p>