9. When fidget spinners were first introduced in the market, Leonard decided to sell them in his toy shop. The number of spinners he sold every week is modeled by \( N(w)= \) \( -w^{2}+10 w \), where \( w \) is the number of weeks starting the first week of February and \( N(w) \) is the number of spinners sold each week in hundreds. One particular week, Leonard realized that his sales were greater than the sales in any other week. What is this maximum number of sales? A. -2500 B. 500 C. 1000 D. 2500 E. 5000

To find the maximum number of fidget spinners sold, we need to analyze the function \( N(w) = -w^2 + 10w \). This is a quadratic function that opens downwards (since the coefficient of \( w^2 \) is negative), which means it has a maximum point. The maximum value of a quadratic function in the form \( ax^2 + bx + c \) occurs at \( w = -\frac{b}{2a} \). In our case, \( a = -1 \) and \( b = 10 \). Calculating the value of \( w \) at which the maximum occurs: \[ w = -\frac{10}{2 \times -1} = \frac{10}{2} = 5 \] Now, we substitute \( w = 5 \) back into the function \( N(w) \) to find the maximum number of spinners sold: \[ N(5) = -5^2 + 10 \times 5 \] \[ N(5) = -25 + 50 \] \[ N(5) = 25 \] Since \( N(w) \) represents the number of spinners sold in hundreds, the actual number of spinners sold is: \[ N(5) = 25 \times 100 = 2500 \] Thus, the maximum number of sales is: \[ \boxed{2500} \]