Couple of maths questions (experts on level 5).

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1

<ol>
<li><p>In the xy-coordinate plane, the graph of y = -x^2 + 9 intersects line L at (p,5) and (t,-7). What is the least possible value of the slope of line L?</p></li>
<li><p>If r+t / r-t = 5/2, what is the value of r/t?</p></li>
<li><p>If (3x10^n) + (5x10^4) = (5.03 x 10^4), what is the value of n?
PS. I was able to do this one using the shift+solve function in the calculator, however, I don't appreciate that kind of dependency on calculators, so I'd like an algebraic approach to this one please. </p></li>
</ol>

<p>Anything is appreciated.</p>

2

<ol>
<li>Plug in the points:</li>
</ol>

<p>(p,5) 5=-p^2 + 9. So -4=-p^2. So 4=p^2. So p=2 or p=-2</p>

<p>(t,-7) -7=-t^2 + 9. So -16=-t^2. So 16=t^2. So t=4 or t=-4.</p>

<p>The slope of the line is (5+7)/(p-t) = 12/(p-t). </p>

<p>This will be small when he denominator is big. We make the denominator as big as possible by choosing the biggest value for p, and the smallest value for t. So choose p=2, t=-4.</p>

<p>Remark: If you’re unsure about which values will make the expression least, just try all 4 possibilities and choose the smallest answer.</p>

<p>Thus the slope is 12/(2+4) = 12/6=2.</p>

3

<p>1) Is the minimum slope -2? Ahh … well DrSteve beat me to #1. I see where he’s going (the same way I went :p). </p>

<p>2) DrSteve is correct. It’s not 1. </p>

<p>3) n = 2. </p>

<p>Tell me if I’m right, and then I’ll elaborate :).</p>

4

<ol>
<li>Cross multiply:</li>
</ol>

<p>r+t / r-t = 5/2</p>

<p>2(r+t) = 5(r-t)</p>

<p>2r+2t = 5r-5t</p>

<p>7t=3r</p>

<p>r/t=7/3</p>

5

<ol>
<li>You can do these computations in your calculator:</li>
</ol>

<p>(3x10^n) + (5x10^4) = (5.03 x 10^4)</p>

<p>3x10^n + 50,000 = 50300</p>

<p>3x10^n = 300</p>

<p>10^n = 100</p>

<p>So n = 2</p>

6

<p>Awesome, DrSteve, thank you for everything. It seems so simple now.</p>

<p>Thank you, IceQube, you too for your contribution.</p>

<p>:-)</p>