Problem 8 - Technical Mathematics with Calculus

Question

What will it cost to cement the floor of a cellar \(25.3 \mathrm{ft}\) long and \(18.4 \mathrm{ft}\) wide, at \(\$ 3.50\) per square foot?

Step-by-Step Solution

Answer

The total cost to cement the cellar floor is \$1,642.24.

Step 1: Calculate the area of the cellar floor

To determine the area of the floor that needs to be cemented, multiply the length by the width. The formula to use is: Area = Length × Width.

Step 2: Multiply the area by the cost per square foot

Once the area is calculated, multiply it by the cost per square foot to find the total cost. Use the formula: Total Cost = Area × Cost per Square Foot.

Step 3: Calculate the total cost

Insert the values into the formula to get the total cost. Ensure to perform correct multiplication taking into account the unit of measurement and the currency.

Key Concepts

Calculating Area

Understanding how to calculate the area of a space is fundamental in many aspects of life, from interior decorating to construction. In our exercise, we’re presented with the task of cementing a cellar floor, which requires the knowledge of how much space we’re dealing with. Calculating area is straightforward: it's the product of length and width. Think of it as finding out how many square tiles we could fit into a rectangular space, with each tile measuring 1 foot on each side.

In mathematical terms, for a rectangle, the formula is \( \text{Area} = \text{Length} \times \text{Width} \). When we apply this to the given measurements of the cellar, \(25.3\text{ ft} \times 18.4\text{ ft}\), we can figure out the total square footage. This calculation forms the foundation for later determining the cost of the project.

Multiplication of Decimals

When dealing with measurements and costs, we often have to multiply decimal numbers. This process can be tricky, but with the right approach, it becomes manageable. The key is to align the decimal points and act as if you're multiplying whole numbers. After the calculations, count the total number of decimal places in both the multiplicands and place the decimal in the result accordingly.

For our cellar example, multiplying the area, \(25.3 \times 18.4\), involves two decimal numbers. We treat them like whole numbers to compute (253 \times 184) and then correct our answer by putting the decimal point in the right place in the final step, considering the original positions of the decimal points in both numbers. This gives us the correct area in square footage, which can then be multiplied by the cost per square foot to find the total price.

Units of Measurement

Every calculation involving physical quantities must take units of measurement into account. In the United States, we often use feet for measuring length and square feet for area. It's crucial to use consistent units when performing calculations to avoid errors. For costs, we deal with currency, like dollars and cents, which require the same attention to detail regarding units.

In our cellar problem, all measurements are given in feet, and the cost is provided per square foot, ensuring consistency. We calculate the total area in square feet and then multiply it by the cost in dollars per square foot to get the total price in dollars. Mixing units, such as using meters and feet together without conversion, would result in incorrect calculations, rendering our efforts useless. Therefore, a deep understanding of units and the necessity for consistent use is vital in problem-solving.