Problem 110 - Algebra for College Students

Question

A gravel dealer charges \(\$ 50\) plus \(\$ 30\) per cubic yard for delivering a truckload of gravel. Express the total cost \(C(n)\) in dollars as a function of the number of cubic yards delivered \(n .\) Find \(C(12)\).

Step-by-Step Solution

Answer

The cost for 12 cubic yards is \$410\.

Step 1: Define the cost function

First, identify the fixed and variable costs. The dealer charges a fixed delivery fee of \$ 50\ and \$ 30\ per cubic yard of gravel. Therefore, the total cost function can be expressed as: \[C(n) = 50 + 30n\]

Step 2: Substitute the given value of cubic yards

Next, substitute \(n = 12\) cubic yards into the cost function to find the total cost.\[C(12) = 50 + 30 \times 12\]

Step 3: Calculate the total cost

Now, calculate the total cost by performing the substitution:\[C(12) = 50 + 360 = 410\]

Key Concepts

Understanding Fixed Cost

Fixed costs are expenses that do not change regardless of how much you produce or sell. In the provided exercise, the gravel dealer charges a fixed delivery fee. This fee remains constant at \(50, no matter how many cubic yards of gravel are delivered.

Fixed costs are crucial to consider because they are expenses that must be covered before you can start making a profit. They are predictable and stable, making budgeting easier.

• Fixed cost in the problem: \)50
• Does not vary with quantity delivered

Exploring Variable Costs

Variable costs depend on the amount of goods or services produced. In the exercise, the cost per cubic yard of gravel is \(30. This means if you order more gravel, the variable part of your cost increases proportionally.

It's important to understand variable costs because they fluctuate with production levels. Unlike fixed costs, variable costs rise as you produce or deliver more, impacting total expenses directly.

• Variable cost in the problem: \)30 per cubic yard
• Changes based on the number of cubic yards (n)

Function Substitution Explained

Function substitution involves replacing a variable in a mathematical expression or function with a specific value. In our exercise, we need to substitute the number of cubic yards, n, with a specific value to calculate the total cost.

For instance, substituting n = 12 in the cost function allows us to find the cost for 12 cubic yards of gravel. This step is demonstrated when we replace n with 12 in the equation: \[ C(12) = 50 + 30 \times 12 \]

• Substitute n with a given value
• Calculate based on the substituted values

Algebraic Expressions

Algebraic expressions contain variables, numbers, and operations (e.g., addition, multiplication). Understanding them is key to manipulating and solving equations.

In the provided problem, the total cost is expressed as an algebraic expression: \[ C(n) = 50 + 30n \ \] This expression combines fixed costs and variable costs. When we solve algebraic expressions, we follow order of operations: parentheses, exponents, multiplication, division, addition, and subtraction (PEMDAS).

• Our function: 50 (fixed cost) + 30n (variable cost)
• Correctly substituting and calculating values