Problem 29 - Applied Physics

Question

How many cubic yards of concrete are needed to pour a patio \(12.0 \mathrm{ft} \times 20.0 \mathrm{ft}\) and \(6.00\) in. thick?

Step-by-Step Solution

Answer

4.44 cubic yards of concrete are needed.

Step 1 - Convert inches to feet

Since the thickness is given in inches, convert it to feet by using the conversion factor: 1 foot = 12 inches. So, \(6\) inches can be converted to feet by dividing by \(12\). Thus, the thickness is \(\frac{6}{12} = 0.5\) feet.

Step 2 - Calculate Volume in Cubic Feet

The volume is found by multiplying the length, width, and height (thickness) of the patio. Use the dimensions given: 12 feet (length), 20 feet (width), and \(0.5\) feet (thickness). The formula for volume is \(\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\). Thus, the volume is \(12 \times 20 \times 0.5 = 120\) cubic feet.

Step 3 - Convert Cubic Feet to Cubic Yards

Since there are \(27\) cubic feet in a cubic yard, convert the volume from cubic feet to cubic yards by dividing the number of cubic feet by \(27\). Thus, \(\frac{120}{27} \approx 4.44\) cubic yards.

Key Concepts

Unit Conversion

Unit conversion is a crucial step when dealing with different measurements. To ensure accurate calculations, it's necessary to convert all units to a consistent format before starting any mathematical operations. In this exercise, we need to convert measurements of thickness from inches to feet. The conversion factor between inches and feet is

• 1 foot = 12 inches

To convert 6 inches into feet, you divide by 12:

• 6 inches = \(\frac{6}{12}=0.5\) feet

This step ensures that all dimensions are in the same unit, making volume calculations straightforward.

Cubic Yards

Cubic yards are often used in construction for measuring large quantities of material like concrete. It is a unit of volume measurement equal to the volume of a cube with sides each one yard (3 feet) long. To convert from cubic feet, a different unit, into cubic yards, you use the fact that:

• 1 cubic yard = 27 cubic feet

When solving the problem, converting the calculated volume of 120 cubic feet into cubic yards involves dividing by 27:

• \(\frac{120}{27} \approx 4.44\) cubic yards

Understanding this conversion is essential for making precise material estimates, ensuring you buy the right amount of concrete.

Mathematical Formulas

Mathematical formulas provide the foundation for solving geometry and volume-related problems. When calculating volume, especially for regular geometric shapes like a rectangular patio, the formula is relatively simple:

• Volume = Length \(\times\) Width \(\times\) Height (or Thickness)

In this scenario, we are calculating the volume in cubic feet using the dimensions:

• Length = 12 feet
• Width = 20 feet
• Thickness = 0.5 feet
• \(\text{Volume} = 12 \times 20 \times 0.5 = 120\) cubic feet

Familiarity with such formulas and their application is key to accurate volume computation.

Concrete Volume

Understanding concrete volume is vital in construction projects like pouring a patio. Concrete is typically measured directly in cubic yards for larger projects. This involves knowing:

• The exact dimensions of the area to be poured
• How to perform unit conversions and use volume formulas

The exercise explores calculating the volume of concrete needed for a patio with these steps:

1. Convert all measurements to the same unit.
2. Calculate the volume using the formula \(\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}\).
3. Convert the volume from cubic feet to cubic yards for purchasing.

Doing so ensures the exact amount of concrete is prepared, reducing waste and costs.