SOLUTION: An appliance service company is located centrally in roughly square area x miles on a side. It charges $27 per call, not including parts and labor, and travel cost is figured at $1

Question 1195967: An appliance service company is located centrally in roughly square area x miles on a side. It charges $27 per call, not including parts and labor, and travel cost is figured at $1.50 per mile. The average distance traveled per call is 1.2x miles. In a month the average number of calls per square mile of service area is 30.
a)What should x be if net travel income (which excludes parts and labor) is to be maximized?
b)What lot size yields minimum cost?

Answer by proyaop(69) (Show Source): You can put this solution on YOUR website!
**a) Determine the Optimal Service Area Size (x) for Maximum Net Travel Income**
* **Net Travel Income:**
* Net Travel Income = (Revenue per call - Travel cost per call) * (Total calls)
* Net Travel Income = ($27 - $1.80x) * (30x²)
* Net Travel Income = 810x² - 54x³
* **Find the derivative of the Net Travel Income function:**
* d(Net Travel Income)/dx = 1620x - 162x²
* **Set the derivative equal to zero to find critical points:**
* 1620x - 162x² = 0
* 162x(10 - x) = 0
* **Solve for x:**
* x = 0 (This is a trivial solution)
* x = 10
* **To verify that x = 10 maximizes net travel income:**
* **Second Derivative Test:**
* d²(Net Travel Income)/dx² = 1620 - 324x
* At x = 10, d²(Net Travel Income)/dx² = 1620 - 324(10) = -1620
* Since the second derivative is negative at x = 10, it indicates a maximum.
* **Therefore, to maximize net travel income, the service area should have a side length (x) of 10 miles.**
**b) Finding the Lot Size for Minimum Cost**
* **The question about "lot size" is not directly related to the given information about the service area.**
* Lot size typically refers to the quantity of goods ordered or produced in a single production run.
* **To determine the lot size that yields minimum cost, you would need additional information:**
* **Ordering costs:** Costs associated with placing an order (e.g., administrative costs, transportation costs).
* **Holding costs:** Costs associated with storing inventory (e.g., storage space, insurance, spoilage).
* **Demand:** The rate at which the goods are used.
* **Common inventory management models (like the Economic Order Quantity - EOQ) can be used to determine the optimal lot size that minimizes the sum of ordering and holding costs.**
**In summary:**
* To maximize net travel income, the service area should have a side length of 10 miles.
* The question about lot size requires additional information to be answered.
I hope this comprehensive explanation is helpful!

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