Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Please answer PART B. Show steps clearly and no cursive if possible. 

### WA-4.1: Optimal Time for Copy Machine Overhaul **Scenario:** Imagine you are running a copy shop with a single copy machine used 24 hours daily. Over time, the machine wears out, requiring periodic replacement with a new one. This guide focuses on finding the optimal time between overhauls, aiming to achieve the minimum monthly cost. **Cost Considerations:** 1. **Fixed cost:** The price of replacing the old copy machine. 2. **Monthly depreciation:** The machine’s value depreciation every month, which increases as the machine gets older. **Tasks:** **(a)** **Given:** The machine's depreciation rate is \( f(s) = \frac{100}{\sqrt{s}} \) dollars per month, where \( s \) is the machine’s age in months. **Task:** Find the total depreciation incurred during the first \( t \) months after an overhaul. The answer should be a function of \( t \). **(b)** **Given:** The price of a new machine is \( A = \$3,000 \). **Task:** Calculate the total depreciation during the interval \([0, t]\) and add it to the fixed cost to get the total cost. **Graph:** Sketch this total cost function. Use "month" for the horizontal axis and "total cost" for the vertical axis. Label the intercept with a specific dollar amount and ensure the graph shows the correct concavity. **(c)** **Task:** From a copy shop owner’s perspective, explain why "Total Cost" is not the main factor to consider. Identify what should be minimized. **(d)** **Given:** Let \( C \) denote the **Average Monthly Cost** \( C(t) \). **Task:** Define and write the average cost function \( C = C(t) \) for \( t \geq 0 \). This function represents the average monthly costs over the interval \([0, t]\). Refer to the textbook for the exact function definition if needed. **(e)** **Task:** Determine the optimal "stopping" time \( T \) to minimize the function \( C(t) \). 1. **Critical Number:** Find the exact value of \( T \). 2. **Behavior of \( C \):** Show \( C \) is decreasing on \( 0 < t < T \) and increasing on \( T
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Transcribed Image Text:### WA-4.1: Optimal Time for Copy Machine Overhaul **Scenario:** Imagine you are running a copy shop with a single copy machine used 24 hours daily. Over time, the machine wears out, requiring periodic replacement with a new one. This guide focuses on finding the optimal time between overhauls, aiming to achieve the minimum monthly cost. **Cost Considerations:** 1. **Fixed cost:** The price of replacing the old copy machine. 2. **Monthly depreciation:** The machine’s value depreciation every month, which increases as the machine gets older. **Tasks:** **(a)** **Given:** The machine's depreciation rate is \( f(s) = \frac{100}{\sqrt{s}} \) dollars per month, where \( s \) is the machine’s age in months. **Task:** Find the total depreciation incurred during the first \( t \) months after an overhaul. The answer should be a function of \( t \). **(b)** **Given:** The price of a new machine is \( A = \$3,000 \). **Task:** Calculate the total depreciation during the interval \([0, t]\) and add it to the fixed cost to get the total cost. **Graph:** Sketch this total cost function. Use "month" for the horizontal axis and "total cost" for the vertical axis. Label the intercept with a specific dollar amount and ensure the graph shows the correct concavity. **(c)** **Task:** From a copy shop owner’s perspective, explain why "Total Cost" is not the main factor to consider. Identify what should be minimized. **(d)** **Given:** Let \( C \) denote the **Average Monthly Cost** \( C(t) \). **Task:** Define and write the average cost function \( C = C(t) \) for \( t \geq 0 \). This function represents the average monthly costs over the interval \([0, t]\). Refer to the textbook for the exact function definition if needed. **(e)** **Task:** Determine the optimal "stopping" time \( T \) to minimize the function \( C(t) \). 1. **Critical Number:** Find the exact value of \( T \). 2. **Behavior of \( C \):** Show \( C \) is decreasing on \( 0 < t < T \) and increasing on \( T
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