To solve the equation \( 9x^3 + 1 = 8112 \), we start by isolating \( 9x^3 \):
\[ 9x^3 = 8112 - 1 \] \[ 9x^3 = 8111 \]
Next, we divide both sides by 9:
\[ x^3 = \frac{8111}{9} \]
Now, calculate \( \frac{8111}{9} \):
\[ \frac{8111}{9} \approx 901.2222 \]
To find \( x \), we take the cube root of \( x^3 \):
\[ x = \sqrt[3]{\frac{8111}{9}} \approx \sqrt[3]{901.2222} \]
Calculating the cube root gives us an approximate value for \( x \):
\[ x \approx 9.65 \]
Since none of the provided options (0, no solution, 3, 9) match this value, the correct choice would be "no solution." Therefore, the equation \( 9x^3 + 1 = 8112 \) does not have a solution in the options provided as they do not correspond to the calculated value of \( x \).