A few BB2 math questions...

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1

<p>Page 527, #8:</p>

<p>(x - 8)(x - k) = x^2 - 5kx + m
In the equation above, k and m are constants. If the equation is true for all values of x, what is the value of m?</p>

<hr>

<p>Page 548, #16:</p>

<p>A cube with volume 8 cm is inscribed in a sphere so that each vertex of the cube touches the sphere. What is the length of the diameter, in cm, of the sphere?</p>

2

<ol>
<li><p>(x-8)<em>(x-k)= x^2-8x-kx+k8 = x^2-5kx+m. We know that k</em>8=m (because both are the only constants). Also, -8x-kx=-5kx (because these are the only terms with x’s on either side. dividing by x, -8-k=-5k, -8=-4k, k=2. So, m=8<em>k (from above), and plugging in k=2 gives m=8</em>2=16.</p></li>
<li><p><a href=“http://www.ul.ie/~cahird/polyhedronmode/Sphere_cube.gif[/url]”>http://www.ul.ie/~cahird/polyhedronmode/Sphere_cube.gif&lt;/a&gt;. Here’s a picture of what this looks like. Anyways, you want to find the quickest way to get the radius/diameter of the sphere. You want to figure what part of the cube represents the sphere’s radius or diameter. With a bit of visualization, hopefully you see that the diagonal of the cube represents the diameter of the sphere (look at the link to confirm). So, we should find the length of the cube’s diagonal. Remember, the cube is 3-D, and the side length is 2cm because the volume is 8cm^3. So, the way to find the diagonal is to first find the diagonal length of the cube’s base (which is the red segment in this link <a href=“http://upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/250px-Cube_diagonals.svg.png[/url]”>http://upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cube_diagonals.svg/250px-Cube_diagonals.svg.png&lt;/a&gt;). By the pythagoreom theorem, this equals (2^2+2^2)^(1/2)=8^(1/2). Now, we have to project this 2-d diagonal into 3-d (the new diagonal–the main one–will be the blue segment in the same link). The base of the new right triangle is 8^(1/2), the height is 2 (just a side length), so the hypotenuse is ((8^(1/2))^2+2^2)^(1/2)=(8+4)^(1/2)=12^(1/2)=root12=(root4)<em>(root3)=2</em>root3. So, the diameter is 2*root(3) cm. Right?</p></li>
</ol>

3

<p>Thanks; I understand it now.</p>

<p>Any idea for my second question?</p>

4

<p>I don’t have paper handy to solve it, but the diameter of the sphere should be equal to the diagonal of the cube.</p>

5

<p>(x - 8)(x - k) = x^2 - 5kx + m</p>

<p>Alternate way:
For x=0
8k = m</p>

<p>For x=8
0 = 64 - (5)8k + m
0 = 64 - 5m + m
m = 16</p>