1. Use the basic resistance formula

For a conductor, $R = \rho \frac{\ell}{A},$ where:

• $\rho$ = resistivity,
• $\ell$ = length of current path,
• $A$ = cross-sectional area perpendicular to current.

2. Identify geometry of the sheet

The sheet is:

• square of side $L$,
• thickness $t$,
• resistance is measured between two opposite faces (the shaded side faces).

So current flows from one side face to the opposite side face across the square, i.e. over a distance $\ell = L.$

3. Find cross-sectional area

The area perpendicular to current is the product of:

• thickness $t$,
• width of the sheet $L$.

Hence, $A = Lt.$

4. Compute resistance

Substitute into $R = \rho \frac{\ell}{A}:$

$R = \rho \frac{L}{Lt} = \frac{\rho}{t}.$

5. Analyze dependence

Thus,

• $R$ is independent of $L$,
• $R$ is inversely proportional to $t$.

So among the options:

• A: directly proportional to $L$ $\to$ false
• B: directly proportional to $t$ $\to$ false
• C: independent of $L$ $\to$ true
• D: independent of $t$ $\to$ false

6. Final answer

The correct option is: $\boxed{\text{C}}$

7. Comparison with stored answer

Stored correct answer = C, which matches the derived result.