To find the maximum profit from the quadratic function \( P(l) = -4l^2 + 32l - 52 \), we can determine the vertex of the parabola represented by this quadratic equation. The vertex will give us the maximum profit because the coefficient of \( l^2 \) is negative, indicating that the parabola opens downwards.
The formula for the \( l \)-coordinate of the vertex of a quadratic function in the form \( P(l) = al^2 + bl + c \) is given by:
\[ l = -\frac{b}{2a} \]
In this case, \( a = -4 \) and \( b = 32 \).
Now we can plug in the values for \( a \) and \( b \):
\[ l = -\frac{32}{2 \times -4} = -\frac{32}{-8} = 4 \]
Now that we have \( l = 4 \), we can substitute this value back into the profit function \( P(l) \) to find the maximum profit:
\[ P(4) = -4(4)^2 + 32(4) - 52 \] \[ = -4(16) + 128 - 52 \] \[ = -64 + 128 - 52 \] \[ = 64 - 52 \] \[ = 12 \]
Therefore, the maximum profit that can be made is \( 12 \) thousand dollars.
So, the final answer is:
\[ \text{Maximum Profit} = 12 \text{ thousand dollars.} \]