Question 1152808
<br>
Let the dimensions of the prism be a, b, and c.  Then<br>
(1) The total length of all the edges is 40:
{{{4(a+b+c) = 40}}}  -->  {{{a+b+c = 10}}}<br>
(2) The total surface area is 48:
{{{2ab+2ac+2bc = 48}}}<br>
The length we are looking for is the length of the space diagonal of the prism, which is<br>
{{{sqrt(a^2+b^2+c^2)}}}<br>
Square equation (1) and substitute equation (2):<br>
{{{(a+b+c)^2 = 100 = a^2+b^2+c^2+2ab+2ac+2bc}}}
{{{100 = a^2+b^2+c^2+48}}}
{{{a^2+b^2+c^2 = 100-48 = 52}}}
{{{sqrt(a^2+b^2+b^2) = sqrt(52) = 2sqrt(13)}}}<br>
The length of the diagonal is 2*sqrt(13).<br>