Triangular-based pyramid $A P Q R$ can be thought of as having triangular base $\triangle A P Q$ and height $A R$. Since this pyramid is built at a vertex of the cube, then $\triangle A P Q$ is right-angled at $A$ and $A R$ is perpendicular to the base. The area of $\triangle A P Q$ is $\frac{1}{2} \times A P \times A Q=\frac{1}{2} x(x+1)$. The height of the pyramid is $\frac{x+1}{2 x}$. Thus, the volume of the pyramid is $\frac{1}{3} \times \frac{1}{2} x(x+1) \times \frac{x+1}{2 x}$ which equals $\frac{(x+1)^{2}}{12}$. Since the cube has edge length 100, its volume is $100^{3}$ or 1000000. Now, $1 \%$ of 1000000 is $\frac{1}{100}$ of 1000000 or 10000. Thus, $0.01 \%$ of 1000000 is $\frac{1}{100}$ of 10000 or 100. This tells us that $0.04 \%$ of 1000000 is 400, and $0.08 \%$ of 1000000 is 800. We want to determine the number of integers $x$ for which $\frac{(x+1)^{2}}{12}$ is between 400 and 800. This is equivalent to determining the number of integers $x$ for which $(x+1)^{2}$ is between $12 \times 400=4800$ and $12 \times 800=9600$. Since $\sqrt{4800} \approx 69.28$ and $\sqrt{9600} \approx 97.98$, then the perfect squares between 4800 and 9600 are $70^{2}, 71^{2}, 72^{2}, \ldots, 96^{2}, 97^{2}$. These are the possible values for $(x+1)^{2}$ and so the possible values for $x$ are $69,70,71, \ldots, 95,96$. There are $96-69+1=28$ values for $x$.