Problem 330 - Calculus Volume 1

Question

For the following exercises, consider a limousine that gets \(m(v)=\frac{(120-2 v)}{5}\) milgal at speed \(v,\) the chauffeur costs \(\$ 15 / \mathrm{h},\) and gas is \(\$ 3.5 / \mathrm{gal} .\) Find the cost per mile at speed \(v\)

Step-by-Step Solution

Answer

The cost per mile is \( \frac{17.5}{120-2v} + \frac{15}{v} \).

Step 1: Determine Fuel Efficiency

Start with the given function for mileage per gallon, which is \( m(v) = \frac{(120 - 2v)}{5} \). This function gives the mileage in miles per gallon at a speed \( v \).

Step 2: Calculate Fuel Cost Per Mile

The cost of gas is \( \$3.5/\text{gal} \). We need to find out how many gallons are used per mile, which is the inverse of the mileage function: \( \frac{1}{m(v)} \). Thus, the fuel cost per mile is \( 3.5 \times \frac{5}{120 - 2v} \).

Step 3: Determine Time Spent Per Mile

At speed \( v \), it takes \( \frac{1}{v} \) hours to travel one mile. Since the chauffeur costs \( \$15/\text{hour} \), the cost for the chauffeur per mile is \( \frac{15}{v} \).

Step 4: Calculate Total Cost Per Mile

Add all costs per mile together to get the total cost: \[ \text{Total cost per mile} = \frac{3.5 \times 5}{120 - 2v} + \frac{15}{v} \].

Step 5: Simplify the Total Cost Expression

Simplifying the expression gives the cost function: \( \text{Cost per mile} = \frac{17.5}{120 - 2v} + \frac{15}{v} \).

Key Concepts

Cost Functions

Cost functions are mathematical models used to determine the total cost associated with producing a certain quantity of goods or, as in this case, operating at a certain speed for a limousine. Here, we analyze the cost per mile, which is directly influenced by different variables such as fuel consumption and chauffeur fees. To construct a cost function that reflects these elements, we identify the cost components:

• Fuel Cost: Based on the limousine's mileage efficiency, given by the formula \( m(v) = \frac{(120-2v)}{5} \), and the fuel price, \( \$3.5/\text{gallon} \), we calculate the fuel cost per mile by taking the inverse of the mileage and multiplying by the gas price.
• Chauffeur Cost: The time taken per mile at speed \( v \) results in chauffeur cost being \( \frac{15}{v} \) dollars per mile.

You sum these individual costs to form the cost function, which helps us understand how expenses change with speed.

Optimization

Optimization is about finding the most efficient or effective solution to a problem. In our example, we want to minimize the total cost per mile for the limousine ride. The complete cost function calculates the expenses:\[\text{Cost per mile} = \frac{17.5}{120 - 2v} + \frac{15}{v}\]Our task is to find the speed \( v \) that minimizes this cost.

To optimize, we typically take the derivative of the cost function with respect to \( v \), set it equal to zero, and solve for \( v \). This process involves:

• Identifying critical points where the function’s derivative, representing the rate of change, is zero.
• Evaluating these points to figure out which provides the minimum cost.

Ultimately, optimization ensures that the limousine operates at a speed that provides the least operational cost per mile.

Fuel Efficiency

Fuel efficiency determines how effectively a vehicle converts fuel into distance traveled, usually expressed in miles per gallon. In the limousine problem, fuel efficiency depends on the speed \( v \) and is defined by the function \( m(v) = \frac{120 - 2v}{5} \). As speed changes, so does the efficiency:

• Higher speeds lead to increased fuel consumption resulting in lower mileage, making rides more expensive.
• Lower speeds might improve mileage to an optimal point but could increase chauffeur costs due to the increased travel time per mile.

Understanding fuel efficiency is key when evaluating the operational costs as it helps balance the trade-offs between speed and cost. Efficiently managing these variables can make a significant difference in achieving minimal operational expenses for the limousine.