To determine how many bags of corn will give the trader the desired profit, we need to analyze the profit function given in the question. The profit function is expressed as: \[ P(x) = 10x - x^2 \] where \( P(x) \) represents the profit in naira from selling \( x \) bags of corn. ### Step 1: Understanding the Profit Function The profit function \( P(x) = 10x - x^2 \) is a quadratic equation. Quadratic equations generally take the form: \[ ax^2 + bx + c \] In our case: - \( a = -1 \) (the coefficient of \( x^2 \)) - \( b = 10 \) (the coefficient of \( x \)) - \( c = 0 \) (there is no constant term) ### Step 2: Finding the Maximum Profit Since the coefficient of \( x^2 \) is negative, the parabola opens downwards, which means it has a maximum point (the vertex). The x-coordinate of the vertex of a quadratic function can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting our values for \( a \) and \( b \): \[ x = -\frac{10}{2 \times -1} = \frac{10}{2} = 5 \] ### Step 3: Calculating the Profit at \( x = 5 \) Now, we can calculate the profit when \( x = 5 \): \[ P(5) = 10(5) - (5)^2 \] \[ P(5) = 50 - 25 \] \[ P(5) = 25 \] This means that the maximum profit of 25 naira occurs when the trader sells 5 bags of corn. ### Step 4: Analyzing the Options Now, let's look at the options provided: - A. 4 - B. 5 - C. 6 - D. 7 From our calculations, we found that the maximum profit occurs at \( x = 5 \). Therefore, the correct option is: **B. 5** ### Step 5: Why Other Options Are Incorrect - **Option A (4)**: If we calculate the profit at \( x = 4 \): \[ P(4) = 10(4) - (4)^2 = 40 - 16 = 24 \] This profit is less than the maximum profit of 25 at \( x = 5 \). - **Option C (6)**: If we calculate the profit at \( x = 6 \): \[ P(6) = 10(6) - (6)^2 = 60 - 36 = 24 \] Again, this profit is less than the maximum profit of 25 at \( x = 5 \). - **Option D (7)**: If we calculate the profit at \( x = 7 \): \[ P(7) = 10(7) - (7)^2 = 70 - 49 = 21 \] This profit is also less than the maximum profit of 25 at \( x = 5 \). ### Summary of Key Points - The profit function is a quadratic equation, which can be maximized. - The maximum profit occurs at the vertex of the parabola, calculated using \( x = -\frac{b}{2a} \). - The maximum profit of 25 naira occurs when the trader sells 5 bags of corn. - Other options yield lower profits than the maximum profit at 5 bags. ### Revision Summary - The profit function is \( P(x) = 10x - x^2 \). - The maximum profit occurs at \( x = 5 \) bags. - Profit calculations for other options (4, 6, 7) yield lower profits. - The correct answer is **B. 5** bags.