In the cost function below, \( C(x) \) is the cost of producing \( x \) items. Find the average cost per item when the required number of items is produced \( C(x)=6.4 x+10,500 \) \( \begin{array}{lll}\text { a. } 200 \text { items } & \text { b. } 2000 \text { items } & \text { c. } 5000 \text { items }\end{array} \)

To find the average cost per item, we need to divide the total cost by the number of items produced. Given: \[ C(x) = 6.4x + 10,500 \] We need to find the average cost per item for the following scenarios: a. 200 items b. 2000 items c. 5000 items Let's calculate the average cost per item for each scenario: a. For 200 items: \[ \text{Average cost per item} = \frac{C(200)}{200} \] b. For 2000 items: \[ \text{Average cost per item} = \frac{C(2000)}{2000} \] c. For 5000 items: \[ \text{Average cost per item} = \frac{C(5000)}{5000} \] Now, let's calculate the average cost per item for each scenario. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6.4\times 200+10500\right)}{200}\) - step1: Remove the parentheses: \(\frac{6.4\times 200+10500}{200}\) - step2: Multiply the numbers: \(\frac{1280+10500}{200}\) - step3: Add the numbers: \(\frac{11780}{200}\) - step4: Reduce the fraction: \(\frac{589}{10}\) a. For 200 items, the average cost per item is $58.9. Now, let's calculate the average cost per item for 2000 items. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6.4\times 2000+10500\right)}{2000}\) - step1: Remove the parentheses: \(\frac{6.4\times 2000+10500}{2000}\) - step2: Multiply the numbers: \(\frac{12800+10500}{2000}\) - step3: Add the numbers: \(\frac{23300}{2000}\) - step4: Reduce the fraction: \(\frac{233}{20}\) b. For 2000 items, the average cost per item is $11.65. Finally, let's calculate the average cost per item for 5000 items. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6.4\times 5000+10500\right)}{5000}\) - step1: Remove the parentheses: \(\frac{6.4\times 5000+10500}{5000}\) - step2: Multiply the numbers: \(\frac{32000+10500}{5000}\) - step3: Add the numbers: \(\frac{42500}{5000}\) - step4: Reduce the fraction: \(\frac{17}{2}\) c. For 5000 items, the average cost per item is $8.5. Therefore, the average cost per item for the given scenarios is: a. 200 items: $58.9 b. 2000 items: $11.65 c. 5000 items: $8.5