Problem 64 - College Algebra

Question

For the following exercises, use a system of linear equations with two variables and two equations to solve. A moving company charges a flat rate of \(\$ 150,\) and an additional \(\$ 5\) for each box. If a taxi service would charge \(\$ 20\) for each box, how many boxes would you need for it to be cheaper to use the moving company, and what would be the total cost?

Step-by-Step Solution

Answer

The moving company is cheaper for more than 10 boxes, costing \( \$205 \) for 11 boxes.

Step 1: Define Variables

Let \( x \) represent the number of boxes. We'll write equations for the cost of each service in terms of \( x \).

Step 2: Write the Cost Equations

For the moving company, the cost equation is \( C_m = 150 + 5x \). For the taxi service, the cost equation is \( C_t = 20x \).

Step 3: Set Up the Inequality

We want to find when the moving company's cost is less than the taxi service's cost: \( 150 + 5x < 20x \).

Step 4: Solve the Inequality

Subtract \( 5x \) from both sides: \( 150 < 15x \). Divide both sides by 15: \( x > 10 \). The moving company is cheaper for more than 10 boxes.

Step 5: Calculate Total Cost for 11 Boxes

Since \( x \) must be greater than 10, try \( x = 11 \). The total cost for the moving company is: \( C_m = 150 + 5(11) = 205 \). The taxi service would cost \( C_t = 20(11) = 220 \). Therefore, the total cost for 11 boxes using the moving company is \( \$205 \).

Key Concepts

Systems of Equations

A system of equations is like a group of statements that hold variables we are trying to solve. When it comes to understanding costs between two services, these systems can help us make a smart decision.

• Each equation in the system represents a different condition or perspective of a problem.
• In this problem, we have two equations, one for the moving company and one for the taxi service.

In our scenario, we have:

• The moving company equation: \( C_m = 150 + 5x \)
• The taxi service equation: \( C_t = 20x \)

By setting these two equations side by side, we can analyze them to find the best choice for the number of boxes to transport.

Inequalities

Inequalities help us compare the solutions provided by systems of equations. They show us when one option is greater, less than, or at least equivalent to another. In this exercise, the goal is to see when using the moving company becomes cheaper than the taxi service.

• We set up the inequality as \( 150 + 5x < 20x \).
• This means we are looking for the point where the moving company's cost is less than the taxi company's cost.

These inequalities need careful solving:

1. Bring similar terms together, like subtracting \(5x\) from both sides, resulting in \(150 < 15x\).
2. Then, divide to solve for \(x\), and find that \( x > 10 \).

So, for any number of boxes greater than 10, the moving company becomes the affordable choice.

Cost Analysis

Cost analysis is about breaking down the total expenses to find the best financial decision. In this scenario, you'll need to consider the fixed and variable costs involved in transport.

• The moving company has a flat rate (\(150) plus a variable cost per box (\)5).
• The taxi service has only a variable cost, charging \(20 per box.

Let's perform a cost analysis to see why choosing the moving company is a good idea with 11 boxes:

• For the moving company: \( C_m = 150 + 5(11) = 205 \). This includes the flat rate plus the cost for each box.
• For the taxi: \( C_t = 20(11) = 220 \). Here, it’s just the box charge multiplied by the number of boxes.

By calculating these costs, you'll notice that the moving company's total cost is lower for 11 boxes, as it equals \)205, compared to the taxi's $220. This shows that proper cost analysis can save money.