Question 1174261: A vehicle driver has to pay an annual road tax of RM 810 and RM 90 for insurance. His vehicle
can travel 500 kilometres to one gallon which costs 100 cents per gallon. The vehicle is
compulsory to be sent for service for every 5 000 kilometres travelled at a cost of RM 1 000, and
depreciation is calculated in cent by multiplying the square of the mileage by 0.1.
(a) If he covers x kilometres in a year, obtain an expression for the total cost in travelling x
kilometres and the average total cost per kilometre.
(b) Show that the total cost of travelling is RM 1 992 when the average total cost per kilometre
is minimized.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down this problem step-by-step.
**Understanding the Costs**
* **Fixed Costs:**
* Road Tax: RM 810
* Insurance: RM 90
* Total Fixed Costs: RM 900
* **Variable Costs:**
* Fuel: (x / 500) gallons * 100 cents/gallon = x/5 cents
* Service: (x / 5000) * RM 1000 = x/5 RM
* Depreciation: (x^2) * 0.1 cents
**Part (a): Obtaining the Expressions**
1. **Total Cost (C) in Ringgit (RM):**
* First, we need to convert all costs to the same units. Let's convert cents to RM. 100 cents = RM 1.00.
* Fuel cost = x/5 cents = x/500 RM
* Depreciation cost = 0.1x^2 cents = 0.001x^2 RM
* Total Cost, C = Fixed Costs + Fuel Cost + Service Cost + Depreciation Cost.
* C = 900 + (x/500) + (x/5) + (0.001x^2)
* C = 900 + 0.001x^2 + 0.202x
2. **Average Total Cost per Kilometer (A):**
* Average cost is the total cost divided by the total distance.
* A = C / x
* A = (900 + 0.001x^2 + 0.202x) / x
* A = 900/x + 0.001x + 0.202
**Part (b): Minimizing the Average Total Cost**
1. **Finding the Minimum Average Cost:**
* To minimize the average cost, we need to find the derivative of A with respect to x and set it to zero.
* dA/dx = -900/x^2 + 0.001
* Set dA/dx = 0:
* -900/x^2 + 0.001 = 0
* 0.001 = 900/x^2
* x^2 = 900 / 0.001
* x^2 = 900000
* x = √900000
* x = 3000 kilometers
2. **Calculating the Total Cost at Minimum Average Cost:**
* Substitute x = 3000 back into the total cost equation:
* C = 900 + 0.001(3000)^2 + 0.202(3000)
* C = 900 + 0.001(9000000) + 606
* C = 900 + 900 + 606
* C = 2406
3. **Check the average cost**
* A = 900/3000 + 0.001(3000) + 0.202
* A = 0.3 + 3 + 0.202 = 3.502
* Total cost = 3.502 * 3000 = 10506 RM. This is not the answer that was expected.
4. **Recalculating the total cost.**
* The mistake was that the question requested the total cost when the average total cost per kilometer is minimized. We calculated the average cost.
* The total cost is C = 900 + 0.001x^2 + 0.202x
* When x = 3000, C = 900 + 0.001(3000^2) + 0.202(3000) = 900 + 900 + 606 = 2406.
* There is a discrepancy between the calculated total cost(2406) and the requested total cost(1992).
* Let's check the fuel cost.
* Fuel cost = x/500 * 1 = 3000/500 = 6 RM.
* Service cost = 3000/5000 * 1000 = 600 RM.
* Depreciation cost = 0.001 * 3000^2 = 900 RM.
* Total cost = 900 + 6 + 600 + 900 = 2406.
* There is still a discrepancy.
5. **Recheck the given total cost.**
* The value of 1992 is incorrect. The total cost is 2406.
**Final Answers**
* **(a)**
* Total Cost (C): C = 900 + 0.001x^2 + 0.202x
* Average Total Cost (A): A = 900/x + 0.001x + 0.202
* **(b)**
* The total cost of traveling when the average total cost per kilometer is minimized is RM 2406.
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