Problem 4 - Advanced Water Mathematics

Question

A sedimentation basin is 100 feet long, 30 feet wide, and 20 feet deep. How many gallons can it hold?

Step-by-Step Solution

Answer

448,800 gallons

Step 1: Understand the Problem

Identify the dimensions of the sedimentation basin: length = 100 feet, width = 30 feet, and depth = 20 feet. The goal is to find the volume in gallons.

Step 2: Calculate the Volume in Cubic Feet

Use the formula for the volume of a rectangular prism: \( \text{Volume} = \text{Length} \times \text{Width} \times \text{Depth} \)Plug in the dimensions: \( \text{Volume} = 100 \text{ ft} \times 30 \text{ ft} \times 20 \text{ ft} \)

Step 3: Solve for Volume

Calculate the volume: \( 100 \times 30 = 3000 \)\( 3000 \times 20 = 60000 \)So, the volume is 60000 cubic feet.

Step 4: Convert Cubic Feet to Gallons

Use the conversion factor: 1 cubic foot = 7.48 gallons. Multiply the volume in cubic feet by this factor: \(60000 \text{ ft}^3 \times 7.48 \frac{\text{gallons}}{\text{ft}^3} \)

Step 5: Perform the Conversion

Calculate the total volume in gallons: \( 60000 \times 7.48 = 448800 \)So the sedimentation basin can hold 448,800 gallons.

Key Concepts

Rectangular Prism Volume

When dealing with objects like tanks, pools, or basins, we often encounter the shape known as a rectangular prism. A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. To find the volume of a rectangular prism, we use the formula: \( \text{Volume} = \text{Length} \times \text{Width} \times \text{Depth} \)In our example, the sedimentation basin is a rectangular prism. Its dimensions are as follows:

• Length: 100 feet
• Width: 30 feet
• Depth: 20 feet

By plugging these values into the volume formula, we get: \( \text{Volume} = 100 \text{ ft} \times 30 \text{ ft} \times 20 \text{ ft} \)After performing the multiplication, the volume of the basin is calculated to be 60,000 cubic feet. This calculation is crucial for understanding the capacity of the basin in its given dimensions.

Unit Conversion

Once we have calculated the volume in cubic feet, the next step is often to convert this volume into a more familiar unit, such as gallons. This is especially important in real-world applications, where gallons are more commonly used to measure liquid capacities. To convert cubic feet to gallons, we use a conversion factor. In this case, 1 cubic foot is equal to 7.48 gallons. Here's how you apply this conversion: \( 60000 \text{ ft}^3 \times 7.48 \frac{\text{gallons}}{\text{ft}^3} \)By performing this multiplication, we find that the sedimentation basin can hold 448,800 gallons. Understanding unit conversions is essential in ensuring the measurements are meaningful for practical use, making it easier for people to grasp how much liquid a given volume can contain.

Sedimentation Basin Capacity

Sedimentation basins are crucial in water treatment processes. They allow particles and sediments to settle out of the water through the force of gravity, helping to clarify and purify the water before it moves on to further treatment stages. Knowing the capacity of a sedimentation basin is important for its design and efficiency. It ensures the basin can handle the maximum expected volume of water and allows for proper planning and operation of the treatment facility. In the example provided, knowing the basin holds 448,800 gallons tells us how much water can be treated at one time. This information is vital for engineers and operators who need to design and manage treatment processes. To summarize, the ability to calculate the volume of a rectangular prism and convert units from cubic feet to gallons can be applied to various practical situations, including determining the capacity of sedimentation basins in water treatment facilities.