<p>the following questions r from one of BB book or some prep book:</p>
<p>1) (x-8)(x-k)=x^2-5kx+m</p>
<p>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>
<p>answer choices are: 8, 16, 24, 32, 40
the answer is: 16 but i don't get it..plz explain</p>
<p>2) A cube with volume 8 cubic meters 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? answer is: 2 rad 3 (approx: 3.46) how?</p>
<p>thnxs =]</p>
<p>1) If the equation is true for all values of x, then the coefficients of x^2, x, and the constant term must be identical on each side. The coefficient of x^2 is 1 on both sides. The coefficient of x is -(8 + k) on the left and -5k on the right. Therefore, 8 + k = 5 k, giving k = 2. The constant term on the left is 8k and on the right, it's m. Therefore, m = 16.</p>
<p>2) First, visualize the cube inside the sphere. Its vertices are all touching the sphere. Therefore, a diagonal of the cube (going from an upper vertex to the opposite lower vertex, through the center of the cube) is equal to the diameter of the sphere (look at a picture if this is not clear). The volume of the cube is 8 m^3. Therefore, one side of the cube is 2 m. To find the length of the diagonal of the cube, first notice that the diagonal of the cube itself makes up a right triangle with one side of the cube and a diagonal across a face of the cube. So first you need to find the length of the diagonal across a face of the cube. Since the side length is 2 m, the diagonal across the face is 2 rad 2 meters (by Pythagoras). Now, find the length of the diagonal of the cube itself: It's the square root of 2^2 + (2 rad 2)^2, in m. So it's the square root of 2^2 (1 + 2) in m, so it's 2 rad 3 m.</p>
<p>Or, after you figure out that the longest diagonal is the same as the shpere's diameter, simply use the formula for the diagonal of a rectangular prism/cuboid: sqrt(l+w+h)</p>
<p>In this case - the length, width, and height are the same. So, you'd get the quare root of 12, which is 3.46</p>
<p>And you didn't even have to go through the details.</p>
<p>PS - I'm not saying OP is wrong, just that it would be quicker to simply use the formula. ;)</p>
<p>Oh, and I used a more direct method for the first one. We were never taught the OP's way, except for complex equations with i.</p>
<p>Since both equations hold true for all values of x, take two values:</p>
<p>For x=9:
(9-8)(9-k) = 81 - 45k + m
=> 9-k = 81 - 45k + m
=> 44k = m + 72</p>
<p>For x=8:
(8-8)(8-k) = 64 - 40k + m
=> 40k = 64 + m</p>
<p>Subtract equation 2 from 1:
440k = 704 + 11m
-440k = 720 + 10m</p>
<p>to get 0 = 16 - m</p>
<p>or m = 16</p>
<p>@anisha1129, I think uttaresh means sqrt(l^2 + w^2 + h^2) in post #4, and not the formula he/she wrote.</p>
<p>Yeah, QuantMech posted the right formula</p>
<p>Oops, yeah. My bad. :( </p>
<p>sqrt(l^2 + w^2 + h^2)</p>
<p>^ thnxs for the correction.</p>