Problem 149 - College Algebra with Modeling and Visualization

Question

Give an example of a quantity occurring in everyday life that can be computed by a function of more than one input. Identify the inputs and the output.

Step-by-Step Solution

Answer

Example: Calculating fuel cost for a trip using distance, fuel efficiency, and fuel price as inputs.

Step 1: Identify a Real-World Example

Let's consider the scenario of calculating the total cost of a road trip, which is a quantity that is determined by a function with multiple inputs.

Step 2: Identify the Inputs

In this example, the inputs are: 1. The distance of the trip (in miles or kilometers). 2. The fuel efficiency of the vehicle (miles per gallon or kilometers per liter). 3. The cost of fuel per unit (dollar per gallon or currency per liter).

Step 3: Identify the Output

The output we want to compute is the total cost of the fuel required for the trip.

Step 4: Formulate the Function

The function to calculate the total cost (\( C \)) can be expressed as:\[C = \left(\frac{\text{Distance}}{\text{Fuel Efficiency}}\right) \times \text{Cost per Unit}\] This function computes how much fuel is needed (distance divided by fuel efficiency) and then multiplies by the cost per unit of fuel.

Step 5: Apply the Formula

Suppose the distance is 300 miles, fuel efficiency is 30 miles per gallon, and the cost is \(3 per gallon. Substituting these values into the formula will give:\[C = \left(\frac{300}{30}\right) \times 3 = 10 \times 3 = 30\] Therefore, the total cost of the trip would be \)30.

Key Concepts

Real-World Examples

Functions of multiple variables are not just confined to the realm of mathematics; they also appear in our everyday life. A perfect real-world example is planning a road trip and calculating the total cost of fuel. Imagine you're organizing a fun trip across the country. To determine how much you will spend on fuel, you take into account several factors: the distance you plan to travel, how fuel-efficient your car is, and the current price of fuel. These factors combine to give you an estimate of the total fuel cost for your trip. By understanding how to use this type of function, you can make more informed decisions and manage your travel budget effectively.

Input-Output Relationship

The input-output relationship in functions helps us understand what needs to be fed into a function and what result we can expect. Inputs are the variables or factors that influence the final value, or output, of the function. In our road trip example, the inputs into our function are:

• The distance of your journey, which tells you how far you will be driving.
• The fuel efficiency of your vehicle, indicating how much distance you cover per unit of fuel.
• The cost of fuel per unit, defining the price you pay for fuel.

Together, these inputs determine the output, which is the total cost of the fuel for the trip. The beauty of this relationship is that by adjusting the inputs (like finding a cheaper fuel option or planning a shorter trip), you can directly affect the outcome, giving you room to optimize your expenses.

Cost Calculation

Calculating costs based on a function of multiple inputs is a crucial skill. To find your total fuel cost (\( C \)), you use the formula:\[C = \left(\frac{\text{Distance}}{\text{Fuel Efficiency}}\right) \times \text{Cost per Unit}\]
This formula means you first determine how many units of fuel you'll need: this is the distance you'll travel divided by your vehicle's fuel efficiency. Then, you multiply this by the cost per unit of fuel.
For example, if you're covering 300 miles, your vehicle gets 30 miles per gallon, and fuel costs are 3 dollars per gallon, your calculation goes as follows:

• Units of Fuel Required: \( \frac{300}{30} = 10 \text{ gallons} \)
• Total Cost: \( 10 \times 3 = 30 \text{ dollars} \)

With this understanding, you can confidently calculate the costs involved in not only trips but various other activities involving multiple factors, allowing effective financial planning.