Question

Remodeling A remodeler charges \(\$ 3.25\) per square foot to refinish a wood floor. How much would it cost to refinish a wood floor that measures 30 feet by 18 feet?

Short answer

The cost is \( \$ 1755 \).

Step-by-step solution

Understand the Problem

We need to find the total cost of refinishing a wood floor given a rate per square foot and dimensions of the floor.

Calculate the Area of the Floor

The floor is rectangular, so we use the formula for the area of a rectangle: \( \, \text{Area} = \text{length} \times \text{width} \, \). Here, the length is 30 feet, and the width is 18 feet. Calculate: \( \, 30 \times 18 = 540 \, \text{square feet} \, \).

Calculate the Total Cost

Now, use the cost per square foot (\( \$ 3.25 \, \text{per square foot} \, \)) to find the total cost: \( \, \text{Total Cost} = \text{Area} \times \text{cost per square foot} \, \). So: \( \, 540 \times 3.25 = 1755 \, \text{dollars} \, \).

Key concepts

Area of a Rectangle

Calculating the area of a rectangle is a fundamental concept in geometry. To find the area, you need to know the length and the width of the rectangle. The formula for the area is:
\[ \text{Area} = \text{length} \times \text{width} \]
For example, if you have a floor that is 30 feet long and 18 feet wide, you calculate the area by multiplying these dimensions together:

• Length: 30 feet
• Width: 18 feet
• Area: 30 feet \( \times \) 18 feet = 540 square feet

Understanding this allows you to know how much space you have within that rectangle. It's a skill you will use in various situations, such as assessing the size of a room or determining the amount of material needed for a project.

Unit Cost Calculation

Unit cost helps us determine pricing based on a unit of measure. In this case, we are dealing with the cost per square foot for refinishing a floor. Knowing the unit cost is crucial for budgeting projects or purchasing materials. Here's how you can calculate the total cost using the unit price:

• Unit cost is given as \( \\( 3.25 \per \text{square foot} \).
• Calculate the area first, here it is 540 square feet.
• Multiply the area by the unit cost: \( 540 \times 3.25 = 1755 \).
• Total cost: \( \\) 1755 \).

This calculation helps you determine the overall budget necessary for the completion of your task. By breaking it down into unit-based costing, you can easily scale your cost estimates for projects of different sizes.

Mathematical Reasoning

Mathematical reasoning is about making sense of the steps needed to arrive at a solution. It involves understanding the problem, breaking it down into manageable parts, and applying formulas and calculations correctly. Here's how reasoning applied to this problem:
1. **Understand the Context**: The problem describes refinishing a floor; we need to find out how much it will cost. 2. **Identify Known Variables**: Recognize that you're given dimensions of the floor and a cost per square foot. 3. **Apply Relevant Formulas**: Use the area formula for a rectangle and multiply by the unit cost. 4. **Verify Your Solution**: Double-check calculations to ensure accuracy, making sure all steps logically lead to a sound conclusion.

• Calculating area shows how much surface you need to cover.
• Using unit cost reveals the total financial investment required.

By mastering reasoning skills, you can systematically approach similar problems and adapt to different scenarios, bolstering your confidence in solving complex mathematical and real-world challenges.