Question 1195986
**a) Determine Net Travel Income**

* **Revenue per call:** $27
* **Travel cost per call:** $1.50/mile * 1.2x miles = $1.80x
* **Net travel income per call:** $27 - $1.80x

* **Number of calls per square mile:** 30 calls/sq mile
* **Total calls:** 30 calls/sq mile * x² sq miles = 30x² calls

* **Total net travel income:** 
    * (Net travel income per call) * (Total calls) 
    * ($27 - $1.80x) * (30x²) 
    * 810x² - 54x³

**b) Maximize Net Travel Income**

* To maximize net travel income, we need to find the value of 'x' that maximizes the function:
    * Net Travel Income = 810x² - 54x³

* **Find the derivative of the 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.**

**Note:**

* This analysis assumes that the number of calls per square mile remains constant regardless of the service area size. 
* In reality, the number of calls per square mile might vary depending on factors like population density and customer distribution within the service area.

I hope this helps! Let me know if you have any other questions.