Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.
1. **Volume of an \(n\)-dimensional sphere:**
   The volume \(V_n\) of a unit \(n\)-dimensional sphere satisfies the recursion:
   \[
   V_n = 2 \int_0^1 V_{n-1} (1 - z^2)^{(n-1)/2} \, dz
   \]
   This recursion comes from integrating out over the last coordinate.

2. **Expectation of \(|z|\):**
   The expectation of \(|z|\) for a uniform distribution over the \(n\)-dimensional sphere can be written as:
   \[
   \mathbb{E}[|z|] = \frac{2}{V_n} \int_0^1 z V_{n-1} (1 - z^2)^{(n-1)/2} \, dz
   \]
   Simplifying, we get:
   \[
   \mathbb{E}[|z|] = \frac{\int_0^1 z (1 - z^2)^{(n-1)/2} \, dz}{\int_0^1 (1 - z^2)^{(n-1)/2} \, dz}
   \]

3. **Approximation of integrals:**
   To approximate the integrals, we use the fact that for large \(n\), the integrand \((1 - z^2)^{(n-1)/2}\) is sharply peaked near \(z = 0\). This suggests that the main contribution to the integral comes from the region where \(z\) is small.

4. **Integral approximation:**
   For large \(n\), we approximate:
   \[
   \int_0^1 (1 - z^2)^{(n-1)/2} \, dz \approx \int_0^1 e^{-(n-1)z^2/2} \, dz
   \]
   Using the substitution \(u = \sqrt{n-1} z\), we get:
   \[
   \int_0^1 e^{-(n-1)z^2/2} \, dz \approx \frac{1}{\sqrt{n-1}} \int_0^{\sqrt{n-1}} e^{-u^2/2} \, du
   \]
   For large \(n\), this integral can be approximated by:
   \[
   \int_0^{\infty} e^{-u^2/2} \, du = \sqrt{\frac{\pi}{2}}
   \]
   Thus,
   \[
   \int_0^1 (1 - z^2)^{(n-1)/2} \, dz \approx \frac{\sqrt{\pi}}{2\sqrt{n-1}}
   \]

5. **Expectation of \(|z|\) approximation:**
   Similarly, we approximate:
   \[
   \int_0^1 z (1 - z^2)^{(n-1)/2} \, dz \approx \int_0^1 z e^{-(n-1)z^2/2} \, dz
   \]
   Using the same substitution \(u = \sqrt{n-1} z\), we get:
   \[
   \int_0^1 z e^{-(n-1)z^2/2} \, dz \approx \frac{1}{n-1} \int_0^{\sqrt{n-1}} u e^{-u^2/2} \, du
   \]
   For large \(n\), this integral can be approximated by:
   \[
   \int_0^{\infty} u e^{-u^2/2} \, du = 1
   \]
   Thus,
   \[
   \int_0^1 z (1 - z^2)^{(n-1)/2} \, dz \approx \frac{1}{n-1}
   \]

6. **Final expectation value:**
   Combining these results, we get:
   \[
   \mathbb{E}[|z|] \approx \frac{\frac{1}{n-1}}{\frac{\sqrt{\pi}}{2\sqrt{n-1}}} = \frac{2}{\sqrt{\pi(n-1)}}
   \]

7. **Distance of the center of gravity:**
   For a hemisphere, the center of gravity is at a distance \(\mathbb{E}[|z|]\) from the center of the sphere. Thus, for \(n = 100\):
   \[
   \mathbb{E}[|z|] \approx \frac{2}{\sqrt{\pi \cdot 99}} \approx \frac{2}{\sqrt{311.017}} \approx \frac{2}{17.64} \approx 0.113
   \]

8. **Relative error:**
   Given a \(10\%\) relative error, the distance can be approximated as:
   \[
   0.113 \pm 0.0113
   \]

The final answer is \(\boxed{0.113 \pm 0.0113}\)