To find the amount of spending on marketing that will achieve the maximum profit, we need to determine the vertex of the quadratic equation represented by \( P = -5x^2 + 1,000x + 5,000 \).
The vertex formula for the x-coordinate of a parabola represented by \( ax^2 + bx + c \) is given by:
\[ x = -\frac{b}{2a} \]
In our equation:
- \( a = -5 \)
- \( b = 1,000 \)
Substituting these values into the vertex formula:
\[ x = -\frac{1,000}{2 \times -5} \] \[ x = -\frac{1,000}{-10} \] \[ x = 100 \]
Thus, the amount spent on marketing to achieve the maximum profit is 100 thousand dollars.