Question 1195967
**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!