Question 1195986: 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 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.
|
|
|
