Answer
$x=3\ in$.
Work Step by Step
Step 1. The volume of the solid can be expressed as the difference between the large rectangular prism and the empty slot; that is
$V=(x+2)(2x+1)(x+5)-3x(x+5)=(x+5)(2x^2+5x+2-3x)=2(x+5)(x^2+x+1)$
Step 2. Letting $V=208$, we have
$x^3+x^2+x+5x^2+5x+5=104$
or
$x^3+6x^2+6x-99=0$
Step 3. Use synthetic division to find a zero $x=3$ as shown in the figure. Then, the equation becomes
$(x-3)(x^2+9x+33)=0$
Step 4. We can find that the only real solution to the above equation is $x=3\ in$.
