Question

Dimensions of a Patio A contractor orders 8 cubic yards of premixed cement, all of which is to be used to pour a patio that will be 4 inches thick. If the length of the patio is specified to be twice the width, what will be the patio dimensions? (1 cubic yard = 27 cubic feet)

Short answer

The patio dimensions are 18 feet wide and 36 feet long.

Step-by-step solution

- Convert cubic yards to cubic feet

First, convert the volume of premixed cement from cubic yards to cubic feet. Since 1 cubic yard is equal to 27 cubic feet, multiply 8 cubic yards by 27:\[ 8 \text{ cubic yards} \times 27 \frac{\text{cubic feet}}{\text{cubic yard}} = 216 \text{ cubic feet} \]

- Convert thickness from inches to feet

The thickness of the patio is given as 4 inches. Convert this to feet by dividing by 12, since there are 12 inches in a foot:\[ 4 \text{ inches} \times \frac{1 \text{ foot}}{12 \text{ inches}} = \frac{1}{3} \text{ feet} \]

- Use the volume formula

Use the volume formula for the patio. The volume is given by length \(l\) times width \(w\) times height \(h\). Given that the volume is 216 cubic feet and the height (thickness) is \( \frac{1}{3} \text{ feet}\), we can set up the equation:\[ l \times w \times \frac{1}{3} = 216 \]

- Simplify the volume equation

Multiply both sides of the equation by 3 to solve for \( l \times w \):\[ l \times w = 216 \times 3 = 648 \]

- Relate length and width

Given that the length is twice the width \( l = 2w \), substitute \(2w\) in place of \(l\) in the equation from Step 4:\[ 2w \times w = 648 \]\[ 2w^2 = 648 \]

- Solve for the width

Solve for \( w \) by dividing both sides of the equation by 2:\[ w^2 = 324 \]Take the square root of both sides:\[ w = \sqrt{324} = 18 \text{ feet} \]

- Solve for the length

Now solve for the length using \( l = 2w \):\[ l = 2 \times 18 = 36 \text{ feet} \]

Key concepts

Volume Conversion

To start solving the problem, we need to convert the volume of the cement from cubic yards to cubic feet. This is essential because it allows us to work with consistent units throughout the calculation. Volume is a measure of the amount of space an object occupies.

• Since 1 cubic yard equals 27 cubic feet, we will use this conversion factor.
• By multiplying 8 cubic yards by 27, we find that 8 cubic yards is equivalent to 216 cubic feet.

This conversion is a crucial first step. It ensures that we are using the same unit of measurement when calculating the patio dimensions later on.

Unit Conversion

In this exercise, we also need to convert the thickness of the patio from inches to feet. Making sure all measurements are in the same unit helps to avoid mistakes.

• The thickness is given as 4 inches, and we know that there are 12 inches in a foot.
• To convert inches to feet, we divide the number of inches by 12. This gives us \( \frac{4}{12} = \frac{1}{3} \) feet.

Now, with the thickness in feet, we can proceed to use it in further calculations. Converting units like this might seem simple, but it's a fundamental skill in various practical applications, especially in construction.

Quadratic Equations

Understanding the relationship between the length and width of the patio involves solving a quadratic equation. These equations are essential for finding areas or volumes when variables multiply each other.

• We start with the equation from our problem: \( l \times w \times \frac{1}{3} = 216 \).
• First, we simplify it to \( l \times w = 648 \) by multiplying both sides by 3.
• We know that the length is twice the width: \( l = 2w \). Substituting \( 2w \) for \( l \), we get \( 2w \times w = 648 \), or \( 2w^2 = 648 \).
• By dividing both sides by 2, we find \( w^2 = 324 \). Taking the square root gives us \( w = 18 \) feet.

With the width found, calculating the length is straightforward. This problem showcases how quadratic equations can simplify complex relationships.

Geometry in Construction

Geometry is a crucial aspect of construction, helping to shape anything from simple patios to entire buildings. Understanding the geometric relationships between different dimensions ensures structures are built correctly.

• The volume formula for a rectangular prism (which a patio resembles) is length \( l \) times width \( w \) times height \( h \).
• In our task, we use this formula to set up our equation: \( l \times w \times \frac{1}{3} = 216 \).
• We break the problem down by converting units and using known relationships, like \( l = 2w \), to solve for the dimensions.

Through these calculations, we find the width is 18 feet and the length is 36 feet. This clear application of geometry helps ensure everything fits together perfectly, an essential practice in any construction project.