Problem 88 - College Algebra with Modeling and Visualization

Question

Discharge of Water The Amazon River discharges water into the Atlantic Occan at an average rate of \(4,200,000\) cubic feet per second, the highest rate of any river in the world. Is this more or less than 1 cubic mile of water per day? Explain your calculations. (Source: The Guinness Book of Reconds I993.) (PICTURE NOT COPY)

Step-by-Step Solution

Answer

The Amazon River discharges more than 1 cubic mile of water per day.

Step 1: Understand the Units

We are given the discharge rate in cubic feet per second \( (ft^3/s) \), and we need to convert this to cubic miles per day. Therefore, we need to know how many seconds are in a day and how many cubic feet are in a cubic mile.

Step 2: Conversion Factors

There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Therefore, the number of seconds in a day is \(24 \times 60 \times 60 = 86,400\) seconds. Also, 1 cubic mile equals \(5280^3\) cubic feet (since 1 mile = 5280 feet).

Step 3: Convert Discharge to Cubic Feet per Day

Multiply the discharge rate by the number of seconds in a day. Thus, the discharge in cubic feet per day is \(4,200,000 \times 86,400 = 362,880,000,000\ ft^3/day.\)

Step 4: Convert Cubic Feet per Day to Cubic Miles per Day

To convert the daily discharge rate from cubic feet to cubic miles, divide by the number of cubic feet in a cubic mile: \(\frac{362,880,000,000}{5280^3} \approx 13.69\) cubic miles per day.

Step 5: Comparison with 1 Cubic Mile

Since the calculated discharge of approximately 13.69 cubic miles per day is more than 1 cubic mile, the Amazon River discharges more than 1 cubic mile of water per day.

Key Concepts

Cubic Feet to Cubic Miles

To solve the problem of comparing the Amazon River's discharge rate in cubic feet to a cubic mile, it's essential to understand the conversion process. One cubic mile is a vast quantity, equivalent to a cube with each side measuring one mile. Since one mile equals 5280 feet, a cubic mile is calculated by taking the cube of this number: \[1 ext{ cubic mile} = 5280^3 ext{ cubic feet}\]By breaking it down, we proceed to multiply 5280 by itself three times, resulting in approximately 147,197,952,000 cubic feet. This large number shows just how much volume a cubic mile represents compared to cubic feet.Understanding this conversion is crucial when working with large-scale volume calculations, such as river discharges and environmental studies. By grasping the relationship between these two units, we can effectively translate various real-world measurements into understandable terms. This foundation allows for clear comparisons in exercises or scientific endeavors.

Discharge Rate

The discharge rate of a river measures the volume of water flowing past a point over a specific period. In our problem, the Amazon River's discharge rate is given as 4,200,000 cubic feet per second. This measurement tells us that every second, this massive river moves that volume of water into the Atlantic Ocean.To gain a deeper perspective, we often need to convert this rate into other time frames, such as daily. By understanding that there are 86,400 seconds in a day (24 hours x 60 minutes x 60 seconds), we can multiply this by the per-second discharge rate:\[4,200,000 ext{ ft}^3/ ext{s} \times 86,400 ext{ s} = 362,880,000,000 ext{ ft}^3/ ext{day}\]This conversion allows us to explore the river's discharge more intuitively and make comparisons, like determining if it surpasses a cubic mile in a day.

Cubic Measurements

Cubic measurements are essential in understanding volume, particularly in applications involving large natural or manmade entities such as rivers, reservoirs, or buildings. Cubic units, such as cubic feet or cubic miles, measure three-dimensional space. When describing how much space something occupies, cubic measurements simplify and standardize the process. Knowing how to convert between different cubic units is vital. For example, understanding that 1 cubic mile is equivalent to 147,197,952,000 cubic feet provides a basis for scaling everyday scenarios to global or regional phenomena like the huge Amazon discharge. By using these measurements proficiently, we gain better insights into environmental science, fluid dynamics, and engineering. They enable us to predict, model, and understand circumstances, which benefit strategic planning, resource management, and infrastructure development. They are the backbone of quantifying physical space in both practical and academic endeavors.