To find the maximum profit of the function \( P(l) = -4l^2 + 32l - 52 \), we can use the vertex formula for a quadratic function, which is given by:
\[ l = -\frac{b}{2a} \]
where \( a \) and \( b \) are the coefficients from the quadratic equation \( ax^2 + bx + c \).
In this case:
- \( a = -4 \)
- \( b = 32 \)
Plugging in these values:
\[ l = -\frac{32}{2 \times -4} = -\frac{32}{-8} = 4 \]
Now, we will find the maximum profit by substituting \( l = 4 \) back into the profit function \( P(l) \):
\[ P(4) = -4(4^2) + 32(4) - 52 \]
Calculating it step by step:
\[ P(4) = -4(16) + 32(4) - 52 \] \[ = -64 + 128 - 52 \] \[ = 128 - 64 - 52 \] \[ = 64 - 52 \] \[ = 12 \]
Thus, the maximum profit is \( P(4) = 12 \) thousand dollars, which is equivalent to \( 12,000 \) dollars.
The answer is \( \boxed{12,000} \).