Problem 40 - Thinking Mathematically

Question

What is the cost of concrete for a walkway that is 15 feet long, 8 feet wide, and 9 inches deep if the concrete costs \(\$ 30\) per cubic yard?

Step-by-Step Solution

Answer

The cost of the concrete for the walkway would be \$100.2.

Step 1: Convert All Measurements to Yards

Firstly, convert all measurements to yards. 1 yard = 3 feet, hence 15 feet = 15/3 = 5 yards and 8 feet = 8/3 = 2.67 yards. Now, convert the depth from inches to yards. 1 yard = 36 inches, hence 9 inches = 9/36 = 0.25 yards.

Step 2: Calculate the Volume of the Walkway

Once the measurements are in the same units, the volume of the walkway is calculated by multiplying the length, width, and depth. Thus, the volume of the walkway is 5 * 2.67 * 0.25 = 3.34 cubic yards.

Step 3: Compute the Cost of Concrete

Lastly, to find the cost of the concrete, multiply the volume of the walkway with the cost per cubic yard. The cost of the concrete will be 3.34 * 30 = \$100.2.

Key Concepts

Volume Calculation

Understanding how to calculate the volume of a three-dimensional object like a walkway is fundamental in geometry. Volume represents the amount of space within an object and is measured in cubic units.

To find the volume, you multiply the object's length, width, and depth (or height). In our example, the walkway has a length of 5 yards, a width of 2.67 yards, and a depth of 0.25 yards.

The formula for volume is:

• Volume = Length \(\times\) Width \(\times\) Depth

By plugging in the measurements:

• Volume = 5 \(\times\) 2.67 \(\times\) 0.25 = 3.34 \text{ cubic yards}

This calculation gives us the total volume of concrete required for the walkway.

Unit Conversion

Unit conversion is a critical skill, especially when dealing with measurements in geometry. Understanding how to convert different units ensures you can accurately calculate other measures like volume.

Let's look at how to convert feet and inches into yards, which is necessary for uniformity in volume calculations.

• 1 yard = 3 feet
• 1 yard = 36 inches

Therefore, for the walkway example, the conversions are as follows:

• 15 feet = \( \frac{15}{3} \) = 5 yards
• 8 feet = \( \frac{8}{3} \) = 2.67 yards
• 9 inches = \( \frac{9}{36} \) = 0.25 yards

Having all dimensions in yards allows you to calculate the accurate volume and, subsequently, the total cost of materials.

Cost Estimation

Estimating costs in projects involving materials such as concrete is a practical application of volume calculations. Once you have the volume, calculating the cost is straightforward.

First, find the total volume (as explained in the volume calculation section) and then multiply this by the cost per cubic unit.

Cost = Volume \(\times\) Cost \, per \, cubic \, yard

For the walkway project, with a volume of 3.34 cubic yards and a cost of \(30 per cubic yard, the total cost is:

• Cost = 3.34 \(\times\) 30 = \\)100.2

In real-world situations, there may be additional considerations like waste factor or delivery fees, but understanding this basic calculation helps set a solid groundwork for cost estimation.