Question

A rectangular channel flows millions of gallons of water through it and dumps into a storage reservoir. The channel is 2 miles long 3 feet wide and 2 feet deep. What is the area of the channel opening?

Short answer

The area of the channel opening is 6 square feet.

Step-by-step solution

- Understand the Problem

The exercise asks for the area of the channel opening. This means calculating the area of the cross-section that carries water.

- Identify the Dimensions

The dimensions of the rectangular channel's cross-section are given as 3 feet wide and 2 feet deep.

- Recall the Area Formula

For a rectangle, the area is calculated as \(\text{Area} = \text{width} \times \text{height}\).

- Calculate the Area

Substitute the values into the formula: \(\text{Area} = 3 \text{ ft} \times 2 \text{ ft} = 6 \text{ square feet}\).

Key concepts

headline of the respective core concept

A rectangular channel is essentially a pathway or bed that directs or holds the flow of water. It's called 'rectangular' because the cross-section of the channel has the shape of a rectangle.
This shape is very common in engineered water systems as it's straightforward to construct and maintain.
Understanding the dimensions and features of a rectangular channel is crucial for various calculations related to water flow.
In our example, the rectangular channel is 2 miles long, 3 feet wide, and 2 feet deep. While the length is important for other calculations, our focus here is on the width and depth to find the cross-sectional area.

headline of the respective core concept

The cross-sectional area is an important concept when dealing with channels or any conduits that transport liquids.
It refers to the area of the slice or cut-out of the shape when viewed from the front. This perpendicular view helps to determine how much space is available for the liquid to flow through.
For a rectangular channel, the cross-section is, naturally, a rectangle.
Knowing the dimensions of the cross-section (width and depth in this case) is key to calculating the area.
The cross-sectional area is crucial in determining the capacity of the channel to carry water and for engineering applications to ensure proper design.

headline of the respective core concept

To calculate the area of a rectangular channel's cross-section, you use the area formula for a rectangle.
The formula is \(\text{Area} = \text{width} \times \text{height}\).
In our example:
\( \text{Area} = 3 \text{ feet} \times 2 \text{ feet} = 6 \text{ square feet} \)
By substituting the given width and depth values into the equation, we find that the cross-sectional area of the channel is 6 square feet.
This formula is universally applicable to any rectangular shape, making it a simple yet powerful tool in various calculations.