To find the maximum profit produced by the function \( P(l) = -4l^2 + 32l - 52 \), we can use the vertex formula for a quadratic function of the form \( ax^2 + bx + c \), where the x-coordinate of the vertex (which gives the maximum or minimum value of the function) is given by:
\[ l = -\frac{b}{2a} \]
In this case, the coefficients are:
- \( a = -4 \)
- \( b = 32 \)
- \( c = -52 \)
Now we can substitute the values of \( a \) and \( b \) into the vertex formula:
\[ l = -\frac{32}{2 \times -4} = -\frac{32}{-8} = 4 \]
Now that we have found \( l = 4 \), we substitute it back into the profit function to find the maximum profit:
\[ P(4) = -4(4^2) + 32(4) - 52 \] \[ = -4(16) + 128 - 52 \] \[ = -64 + 128 - 52 \] \[ = 64 - 52 \] \[ = 12 \]
Since the profit \( P(l) \) is given in thousands, the maximum profit is \( 12,000 \).
Therefore, the answer is: B. $12,000