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Question:
rewrite and Give step-by-step solution with full calcuation and explanation and final answer:
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This is AI generated Response.
Asked by: mayank
Created at: 2025-09-16 18:43:12
Asked by: mayank
Created at: 2025-09-16 18:43:12
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# Step-by-Step Solution
Given:
- **New car cost**: \$10,000
- **Annual operating cost and resale value** given in table
- **A new car can be used for at most 3 years**
- **Objective:** Find a replacement policy over 6 years that minimizes net cost
### Table: Resale Value and Operating Cost
| Age (years) | Resale Value (\$) | Operating Cost (\$) |
|-------------|------------------|---------------------|
| 1 | 7,000 | 300 |
| 2 | 6,000 | 500 |
| 3 | 4,000 | 800 |
| 4 | 3,000 | 1,200 |
| 5 | 2,000 | 1,600 |
| 6 | 1,000 | 2,200 |
### Constraints
- **A car must be replaced after 3 years.**
- We need to plan for **6 years**.
---
## **Step 1: List All Possible Replacement Policies**
Since the car can only be used for a maximum of 3 years, possible policies for 6 years are:
- **[3, 3]**: Replace every 3 years (use new car for 3 years, then another new car for 3 years)
- **[2, 2, 2]**: Replace every 2 years (use new car for 2 years, replace, do this 3 times)
- **[1, 1, 1, 1, 1, 1]**: Replace every year (buy new car each year for 6 years)
Let's calculate the net cost for each.
---
## **Step 2: Calculate Net Cost for Each Policy**
### **Policy 1: Replace Every 3 Years ([3, 3])**
#### **First 3 Years**
- **Purchase Price**: \$10,000
- **Operating Costs**: \$300 (Year 1) + \$500 (Year 2) + \$800 (Year 3) = \$1,600
- **Resale Value after 3 years**: \$4,000
- **Net Cost (First 3 years)**:
\[
\$10,000 - \$4,000 + \$1,600 = \$7,600
\]
#### **Next 3 Years (Repeat the Same Process)**
- **Purchase Price**: \$10,000
- **Operating Costs**: \$300 (Year 4) + \$500 (Year 5) + \$800 (Year 6) = \$1,600
- **Resale Value after 3 years**: \$4,000
- **Net Cost (Next 3 years)**:
\[
\$10,000 - \$4,000 + \$1,600 = \$7,600
\]
#### **Total Net Cost for 6 Years**
\[
\$7,600 + \$7,600 = \$15,200
\]
---
### **Policy 2: Replace Every 2 Years ([2, 2, 2])**
#### **Each 2-Year Cycle**
- **Purchase Price**: \$10,000
- **Operating Costs**: \$300 (Year 1) + \$500 (Year 2) = \$800
- **Resale Value after 2 years**: \$6,000
- **Net Cost per 2 years**:
\[
\$10,000 - \$6,000 + \$800 = \$4,800
\]
#### **Total for 6 Years (3 Cycles)**
\[
3 \times \$4,800 = \$14,400
\]
---
### **Policy 3: Replace Every Year ([1, 1, 1, 1, 1, 1])**
#### **Each Year**
- **Purchase Price**: \$10,000
- **Operating Costs**: \$300
- **Resale Value after 1 year**: \$7,000
- **Net Cost per year**:
\[
\$10,000 - \$7,000 + \$300 = \$3,300
\]
#### **Total for 6 Years**
\[
6 \times \$3,300 = \$19,800
\]
---
## **Step 3: Choose the Minimum Net Cost Policy**
- **Replace every 3 years:** \$15,200
- **Replace every 2 years:** \$14,400 **(lowest)**
- **Replace every year:** \$19,800
---
## **Step 4: Final Answer**
### **Optimal Replacement Policy**
**Replace the car every 2 years.**
### **Minimum Net Cost over 6 Years**
**\$14,400**
---
## **Summary Table**
| Policy | Total Net Cost (\$) |
|----------------|--------------------|
| Every 1 year | 19,800 |
| Every 2 years | 14,400 |
| Every 3 years | 15,200 |
**Final Answer:**
> The minimum net cost is **\$14,400** if you **replace the car every 2 years**.
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