jmc

To find the volume of the box, we multiply the three dimensions: $(x+5)(x-5)(x^2+25)=(x^2-25)(x^2+25)=x^4-625$. We want to find $x$ such that $x^4-625<700$, which simplifies to $x^4<1325$. Taking the fourth root shows us that $x$ is less than $\sqrt[4]{1325}$, which is between 6 and 7 (since $6^4=1296$ while $7^4=2401$). So $x$ could be 1, 2, 3, 4, 5, or 6. However, we see that the width is $x-5$ units, and this must be a positive number, so the only value of $x$ which works is 6. Thus, there is only $\boxed{1}$ possible value of $x$.