Question 1174372: 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! Let's break down this problem step-by-step.
**(a) Total Cost and Average Total Cost**
1. **Fixed Costs:**
* Annual Road Tax: RM 810
* Annual Insurance: RM 90
* Total Fixed Costs: RM 810 + RM 90 = RM 900
2. **Fuel Cost:**
* Distance per gallon: 500 km
* Cost per gallon: 100 cents = RM 1
* Number of gallons needed for x km: x / 500
* Fuel Cost: (x / 500) * 1 = x / 500 RM
3. **Service Cost:**
* Service every 5000 km: x / 5000 services
* Cost per service: RM 1000
* Service Cost: (x / 5000) * 1000 = x / 5 RM
4. **Depreciation Cost:**
* Depreciation per km: 0.1 cents = RM 0.001
* Total depreciation: 0.001 * x² RM
5. **Total Cost (C(x)):**
* C(x) = Fixed Costs + Fuel Cost + Service Cost + Depreciation Cost
* C(x) = 900 + x / 500 + x / 5 + 0.001x²
* C(x) = 900 + 0.002x + 0.2x + 0.001x²
* C(x) = 0.001x² + 0.202x + 900
6. **Average Total Cost per Kilometer (A(x)):**
* A(x) = C(x) / x
* A(x) = (0.001x² + 0.202x + 900) / x
* A(x) = 0.001x + 0.202 + 900 / x
**(b) Minimum Average Total Cost**
1. **Find the Derivative of A(x):**
* A(x) = 0.001x + 0.202 + 900x⁻¹
* A'(x) = 0.001 - 900x⁻²
* A'(x) = 0.001 - 900 / x²
2. **Set A'(x) = 0 to Find Critical Points:**
* 0.001 - 900 / x² = 0
* 0.001 = 900 / x²
* x² = 900 / 0.001
* x² = 900000
* x = √900000 = 948.68 km
3. **Find the Minimum Average Total Cost:**
* Substitute x = 948.68 into A(x):
* A(948.68) = 0.001(948.68) + 0.202 + 900 / 948.68
* A(948.68) ≈ 0.94868 + 0.202 + 0.94868
* A(948.68) ≈ 2.09936 RM/km
4. **Find the Total Cost at x = 948.68 km:**
* C(948.68) = 0.001(948.68)² + 0.202(948.68) + 900
* C(948.68) ≈ 900 + 202 + 900 = 1992 RM
5. **Show that it is a Minimum:**
* Find the second derivative of A(x):
* A'(x) = 0.001 - 900x⁻²
* A''(x) = 1800x⁻³ = 1800 / x³
* Since x > 0, A''(x) > 0. This means the function is concave up, and we have a minimum.
**Conclusion:**
* **(a)** The total cost is C(x) = 0.001x² + 0.202x + 900, and the average total cost is A(x) = 0.001x + 0.202 + 900 / x.
* **(b)** When the average total cost per kilometer is minimized (at x ≈ 948.68 km), the total cost is approximately RM 1992.
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