Problem 56 - Algebra 2

Question

The amount that a mail-order company charges for shipping and handling is given by the function \(c(x)=3+0.15 x\) where \(x\) is the weight in pounds. Find the charge for an 8 -pound order.

Step-by-Step Solution

Answer

The charge for an 8-pound order is $4.20.

Step 1: Identify Values

We need to find the charge for shipping an 8-pound order. In the function \(c(x) = 3 + 0.15x\), the variable \(x\) represents the weight in pounds. Here, \(x = 8\).

Step 2: Substitute into the Function

Substitute \(x = 8\) into the function: \[c(8) = 3 + 0.15 \times 8\].

Step 3: Perform Multiplication

Calculate \(0.15 \times 8\): \[0.15 \times 8 = 1.2\].

Step 4: Add to the Base Cost

Add the result from the multiplication to the base charge: \[3 + 1.2 = 4.2\].

Step 5: Final Charge

The final charge for an 8-pound order is \(c(8) = 4.2\) dollars.

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a particular input. In our example, we use the function \(c(x) = 3 + 0.15x\), which calculates the shipping charge based on the weight of the order in pounds.

To evaluate this function, follow these steps:

• Identify the input value you want to evaluate, which in this case is the weight \(x = 8\) pounds.
• Substitute this value into the function, replacing \(x\) with 8.
• Calculate the resulting expression to find the output, which gives us the shipping charge for the specific weight.

This simple process helps us use functions to easily predict outcomes based on different inputs. Mastering function evaluation allows you to solve real-world problems, like calculating costs, quickly and accurately.

Arithmetic Operations

Arithmetic operations are fundamental mathematical processes such as addition, subtraction, multiplication, and division. These operations are the building blocks for solving equations and evaluating functions, such as our shipping charge example.

In the function \(c(x) = 3 + 0.15x\), we need to perform several arithmetic operations:

• Multiplication: Calculate \(0.15 \times 8\). This operation gives us the additional charge based on the weight.
• Addition: Add the result of the multiplication to the initial base charge of 3 dollars.

Understanding these operations in isolation helps break down complex problems into manageable steps. Each arithmetic operation moves us closer to the final answer, showing how small calculations can lead to solving more complex equations.

Problem Solving

Problem solving involves using logical steps and mathematical operations to find a solution. In our example, the task is to determine the shipping cost for an 8-pound order using the given function. This involves several essential steps:

• Analyze the problem: Understand what the function represents and identify the unknowns and knowns—in this case, the weight and the function itself.
• Plan how to solve it: Recognize the need to substitute the given weight into the function.
• Execute the plan: Perform the necessary calculations, such as multiplication and addition, to find the final shipping cost.
• Reflect on the result: Verify that the solution makes sense in the context of the problem, ensuring that all steps were applied correctly.

Developing problem-solving skills not only aids in mathematics but also builds logical thinking, critical for tackling various challenges in everyday life.