The function P(l) = −4l^2 +32l − 52

To find the maximum profit from the function \( P(l) = -4l^2 + 32l - 52 \), we can identify that this is a quadratic function in the form \( P(l) = ax^2 + bx + c \), where \( a = -4 \), \( b = 32 \), and \( c = -52 \).

The maximum profit occurs at the vertex of the parabola described by the quadratic function. The \( l \)-coordinate of the vertex can be found using the formula:

\[
l = -\frac{b}{2a}
\]

Substituting in the values of \( a \) and \( b \):

\[
l = -\frac{32}{2 \times -4} = -\frac{32}{-8} = 4
\]

Now that we have the value of \( l \) at which the profit is maximized, we substitute \( l = 4 \) back into the profit function \( P(l) \) to find the maximum profit:

\[
P(4) = -4(4^2) + 32(4) - 52
\]
\[
= -4(16) + 128 - 52
\]
\[
= -64 + 128 - 52
\]
\[
= 64 - 52
\]
\[
= 12
\]

Thus, the maximum profit that can be made is **12 thousand dollars**.