Let's break down the problem step by step. ### Problem Details Recap - The first **9400 bottles** fetch a profit of **$6 per bottle**. - For bottles **beyond 9400**, the profit **drops by $0.0004 per additional bottle**. - We need to find: 1. The **maximum profit** (assuming at least 9400 bottles are produced and sold). 2. The **profit per bottle** at this maximum. --- ### 1. Write the Profit Function Let **x** be the total number of bottles produced and sold, where **x ≥ 9400**. #### - For the first 9400 bottles: Profit = \( 9400 \times 6 = \$56,400 \) #### - For bottles beyond 9400 (i.e., for \( x > 9400 \)): - Number of bottles beyond 9400: \( x - 9400 \) - Profit per each bottle beyond 9400: \( 6 - 0.0004(y) \), where \( y \) is the number of bottles above 9400. So, total profit from bottles beyond 9400: \[ \text{Profit from extra bottles} = \sum_{y=1}^{x-9400} [6 - 0.0004y] \] This is an arithmetic series: \[ \text{Sum} = n \cdot a_1 + \frac{n(n-1)}{2}d \] But since the drop is linear, we can use the formula for the sum: \[ \text{Profit from extra bottles} = (x-9400) \cdot 6 - 0.0004 \cdot \frac{(x-9400)(x-9399)}{2} \] #### - Total Profit Function: \[ P(x) = 56,400 + (x-9400) \cdot 6 - 0.0004 \cdot \frac{(x-9400)(x-9399)}{2} \] --- ### 2. Find the Maximum Profit Take the derivative and set to 0 (to find the maximum): Let \( n = x - 9400 \). Then \( x = n + 9400 \): \[ P(n) = 56,400 + 6n - 0.0004 \cdot \frac{n(n+1)}{2} \] \[ P(n) = 56,400 + 6n - 0.0002n(n+1) \] \[ P(n) = 56,400 + 6n - 0.0002n^2 - 0.0002n \] \[ P(n) = 56,400 + (6 - 0.0002)n - 0.0002n^2 \] Set the derivative to zero: \[ P'(n) = 6 - 0.0002 - 0.0004n = 0 \] \[ 0.0004n = 6 - 0.0002 \] \[ 0.0004n = 5.9998 \] \[ n = \frac{5.9998}{0.0004} = 14,999.5 \] So, \[ x = n + 9400 = 14,999.5 + 9400 = 23,399.5 \] Round down to nearest whole bottle: \[ x = 23,399 \] --- ### 3. Maximum Profit Plug \( n = 14,999 \) into the profit function: \[ P(14,999) = 56,400 + 6 \times 14,999 - 0.0002 \times (14,999^2 + 14,999) \] Calculate: - \( 6 \times 14,999 = 89,994 \) - \( 14,999^2 = 224,970,001 \) - \( 224,970,001 + 14,999 = 224,985,000 \) - \( 0.0002 \times 224,985,000 = 44,997 \) So, \[ P(14,999) = 56,400 + 89,994 - 44,997 = 101,397 \] --- ### 4. Profit Per Bottle \[ \text{Profit per bottle} = \frac{101,397}{23,399} \approx \$4.33 \] --- ## **Final Answers:** 1. **Maximum profit:** \(\boxed{101,397}\) 2. **Profit per bottle:** \(\boxed{4.33}\)