Question 1181236
.


            As the problem is worded and presented,  it is  TERRIBLE.


            Nevertheless,  I was able to decipher its hidden meaning.


            So,  I will solve the problem,  but do not ask me how I guessed my interpretation.



<pre>
Let x, y and z be the three dimensions of the prism.

Then we are given these three equations


    xy = 6    (1)

    xz = 9    (2)

    yz = 12   (3)


Multiply these equations. You will get


    (xyz)^2 = 6*9*12 = 648 = (2*3) * (3*3) * (3*4) = 2 * 3^4 * 4 = 2^3 * 3^4


which implies

     xyz = {{{sqrt(648)}}} = {{{sqrt(2^3*3^4)}}} = {{{2*3^2*sqrt(2)}}} = {{{18*sqrt(2)}}}.    (4)



Now       divide equation (4) by equation (1).  You will get  z = {{{(18*sqrt(2))/6}}} = {{{3*sqrt(2)}}}.


Next,     divide equation (4) by equation (2).  You will get  y = {{{(18*sqrt(2))/9}}} = {{{2*sqrt(2)}}}.


Finally,  divide equation (4) by equation (3).  You will get  x = {{{(18*sqrt(2))/12}}} = {{{(3/2)*sqrt(2)}}}.


       +-----------------------------------------------------+
       |   So, at this point we know all the dimensions,     | 
       |   and now I am ready answer the questions.          |
       +-----------------------------------------------------+



(a)  <U>To the nearest tenth of a centimeter, what is the length of the diagonal of the prism?</U>


         The square of the diagonal length is  d^2 = x^2 + y^2 + z^2 = {{{(9/4)*2 + 4*2 + 9*2}}} = 38.5;

         hence, the diagonal is  d = {{{sqrt(38.5)}}} = 6.2 cm  (rounded).      <U>ANSWER</U>



(b)  <U>What is the volume of the prism?</U>


         The volume is  V = xyz = {{{(3/2)*sqrt(2)}}}.{{{2*sqrt(2)}}}.{{{3*sqrt(2)}}} = {{{(3/2)*2*3*2*sqrt(2)}}} = {{{18*sqrt(2)}}} = 25.456 cm^3  (rounded).    <U>ANSWER</U>
</pre>

The problem is solved.


All the questions are answered.


You get a brilliant solution with the careful explanation.



//////////////



Do not forget to post your &nbsp;"THANKS" &nbsp;to me for my teaching.