Question

A construction crew will be installing 2,500 feet of 18 -inch diameter pipe. The width of the trench will be 40 inches and the depth 45 inches. After the pipe has been installed, how many cubic yards of dirt will be needed to backfill the trench? (Assume the trench will be backfilled up to 8 inches from the ground surface.)

Short answer

949.44 cubic yards

Step-by-step solution

- Convert dimensions to feet

First, convert all measurements from inches to feet since the final answer should be in cubic yards (1 yard = 3 feet). The width of the trench is 40 inches, which is \ \ \[\frac{40}{12} = 3.33\] feet. The depth of the trench is 45 inches, which is \ \ \[\frac{45}{12} = 3.75\] feet.

- Calculate the effective depth of the trench

The trench will be backfilled up to 8 inches from the surface. Convert 8 inches to feet: \ \ \[\frac{8}{12} = 0.67\] feet. Subtract this from the total depth of 3.75 feet: \ \ \[3.75 - 0.67 = 3.08\] feet.

- Calculate the volume of the trench

The volume of the trench that needs to be backfilled is given by the formula for the volume of a rectangular prism: \ \ \[ \text{Volume} = \text{length} \times \text{width} \times \text{depth} \]. Here, the length is 2,500 feet, the width is 3.33 feet, and the effective depth is 3.08 feet. Hence, \ \ \[ \text{Volume} = 2500 \times 3.33 \times 3.08 = 25635.0 \text{ cubic feet} \].

- Convert the volume from cubic feet to cubic yards

There are 27 cubic feet in one cubic yard. So, convert the volume obtained in the previous step to cubic yards by dividing by 27: \ \ \[ \text{Volume in cubic yards} = \frac{25635.0}{27} = 949.44 \text{ cubic yards} \].

Key concepts

Unit Conversion

When solving problems involving different units, converting measurements to a common unit is essential. In this problem, we have dimensions given in inches and a requirement to provide the final answer in cubic yards. We start by converting inches to feet because feet are a smaller unit and will simplify our calculations before converting to cubic yards. Here’s how it’s done:

• 1 yard = 3 feet
• 1 foot = 12 inches

To convert from inches to feet:
\( \text{feet} = \frac{\text{inches}}{12} \)
Example: 40 inches to feet: \( \frac{40}{12} = 3.33 \) feet.
Similarly, convert 45 inches to feet: \( \frac{45}{12} = 3.75 \) feet. This step ensures our final volume calculation is accurate and makes it easier to convert into cubic yards later on.

Volume Calculation

Calculating volume is key to determining how much material is needed in this problem. The trench is a rectangular prism, and we use the volume formula for this shape:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{depth} \]
Let's break it down for our trench:

• Length of the trench: 2,500 feet.
• Width of the trench: converted from 40 inches to 3.33 feet.
• Depth of the trench: adjusted depth is 3.08 feet (3.75 feet - 0.67 feet (8 inches converted to feet)).

Using these dimensions:
\[ \text{Volume} = 2500 \times 3.33 \times 3.08 = 25635.0 \text{ cubic feet} \]
This volume calculation gives us the total cubic feet of dirt needed to backfill the trench.

Rectangular Prism

Understanding the trench as a rectangular prism is crucial. A rectangular prism has six faces with opposite faces being equal in area. Its volume is found by multiplying its length, width, and depth. Key characteristics:

• All angles are right angles.
• The opposite faces are parallel and equal.
• It is a 3-dimensional shape.

In our problem, by considering the trench as a rectangular prism, we simplify the volume calculation process.
By using the rectangular prism volume formula:
\[ \text{Volume} = \text{length} \times \text{width} \times \text{depth} \]
we ensure our calculation is systematic and easy to follow. This is an essential reduction to make the problem more manageable and approachable, leading us to accurately determine the amount of backfill needed.