Problem 73 - Precalculus : Building Concepts and Connections

Question

A truck rental company charges a daily rate of \(\$ 15\) plus \(\$ 0.25\) per mile driven. What is the average cost per mile of driving \(x\) miles per day? Use this expression to find the average cost per mile of driving 50 miles per day.

Step-by-Step Solution

Answer

$0.55

Step 1: Identify the Cost Function

The cost function of the truck rental is given as \(c(x) = 15 + 0.25x\), where \(c(x)\) is the total cost of renting the truck and driving x miles in a day.

Step 2: Calculate the Average Cost

The average cost can be calculated by dividing the total cost by the number of miles. So the average cost per mile, \(a(x)\), of driving \(x\) miles per day is given by the formula \(a(x) = c(x)/x = (15 + 0.25x)/x\).

Step 3: Substitute x with 50

To find the average cost per mile of driving 50 miles per day, substitute \(x\) with 50 in the average cost formula \(a(x) = (15 + 0.25*50)/50\).

Step 4: Solve the Equation

Using the formula from step 2, the average cost per mile of driving 50 miles per day is \(a(50) = (15 + 0.25*50)/50 = 0.55\). So, the average cost per mile of driving 50 miles per day is $0.55.

Key Concepts

Cost Function

When we talk about a cost function, we're discussing a mathematical way to represent the cost of a service or product in relation to varying quantities. It's essentially an equation where the output, usually referred to as c(x), describes total cost based on different inputs, denoted by x, which can be things like the number of items produced or, in our case, the number of miles driven.

The cost function for the truck rental in our exercise, c(x) = 15 + 0.25x, combines a fixed cost and a variable cost. The fixed cost ($15) represents a daily rate that does not change regardless of the miles driven, while the variable cost (0.25x) changes depending on how many miles are driven. Understanding how to create and interpret a cost function is vital for making informed financial decisions and for calculating other useful metrics, like the average cost per mile.

Variable Expression

A variable expression in mathematics is an expression that includes variables, such as x, y, or z, along with numbers and operation symbols. It can vary depending on the values substituted for the variables. In our truck rental example, the variable expression within the cost function is 0.25x, which represents the cost per mile driven.

Distinguishing Between Variables and Constants

It's crucial to differentiate between variables and constants. In the expression 0.25x, the number 0.25 is a constant, meaning it's a specific, unchanging amount for every mile driven. Conversely, x is a variable as it represents the number of miles, which can change with every different scenario. By mastering the art of working with variable expressions, one can more easily manipulate and evaluate functions, ultimately helping to solve real-world problems such as calculating costs over various distances.

Evaluating Functions

The process of evaluating functions involves substituting a value for the variable in a function and calculating the result. This is a fundamental skill in algebra that allows us to find out what the output of a function will be for any given input.

In reference to our exercise, we evaluate the average cost function, a(x) = (15 + 0.25x) / x, by replacing x with the specific number of miles driven, which is 50 in this case. After substitution, we perform the arithmetic needed to find the average cost for that distance. This evaluation gives us the precise average cost per mile for a specific day's usage of the truck.

Understanding how to evaluate functions is not just a powerful tool for solving math problems; it is also essential for interpreting relationships in various scientific, economic, and engineering contexts, making this skillset highly applicable in numerous fields.