Question

Since its formation 10,000 years ago, Niagara Falls has eroded upstream a distance of 9.8 miles. Which of the following equations indicates the distance \(D\) that Niagara Falls, continuing at this rate, will erode in the next 22,000 years? A. \(\frac{9.8}{10,000}=\frac{D}{22,000}\) B. \(\frac{9.8}{10,000}=\frac{D}{12,000}\) C. \(D=9.8+\frac{22,000}{10,000}\) D. \(D=9.8 \times \frac{10,000}{22,000}\)

Short answer

The correct equation is \(\frac{9.8}{10,000}=\frac{D}{22,000}\).

Step-by-step solution

Analyze the given information

We know that Niagara Falls has eroded 9.8 miles in 10,000 years. We can think of this as a rate per year. Rate = \(\frac{Distance}{Time}\) So, the rate is \(\frac{9.8 \text{ miles}}{10,000 \text{ years}}\).

Determine the relationship between the given information and the future erosion

We want to find the distance D that Niagara Falls will erode in the next 22,000 years. Since the rate per year remains the same, we can set up a proportion to find the distance D.

Set up a proportion

Let's set up a proportion comparing the distance eroded in 10,000 years to the distance D eroded in 22,000 years: \(\frac{9.8 \text{ miles}}{10,000 \text{ years}} = \frac{D}{22,000 \text{ years}}\) Now, we have to find the correct option among the given choices.

Compare the proportion to the given equation choices

Let's look at each answer choice and compare it to the proportion we set up: A. \(\frac{9.8}{10,000} = \frac{D}{22,000}\) (This matches our proportion) B. \(\frac{9.8}{10,000} = \frac{D}{12,000}\) (The denominator for D should be 22,000, not 12,000) C. \(D = 9.8 + \frac{22,000}{10,000}\) (This is not a proportion) D. \(D = 9.8 \times \frac{10,000}{22,000}\) (This doesn't represent the relationship we're looking for)

Choose the correct equation

Based on our comparison, option A is the only equation that correctly represents the relationship between the given information and the distance D that Niagara Falls will erode in the next 22,000 years: \(\boxed{\frac{9.8}{10,000}=\frac{D}{22,000}}\).

Key concepts

Rate of Change

When discussing the rate of change, it helps to visualize how fast or slow something happens over a period of time. In mathematical terms, the rate of change is the change in one quantity relative to a change in another. In this exercise, the focus is on how far Niagara Falls has eroded over time.

To calculate the erosion rate, we use the formula: \[\text{Rate} = \frac{\text{Distance}}{\text{Time}}\]Here, Niagara Falls eroded 9.8 miles over 10,000 years, giving us a rate of: \[\frac{9.8}{10,000}\] miles per year. This rate helps us determine erosion distance over another span of years. As both time periods share the same condition, the rate becomes a vital tool for future predictions.

Niagara Falls Erosion

Understanding Niagara Falls erosion involves distinguishing how natural forces work over vast periods. Erosion happens when water flow gradually wears away land. For Niagara Falls, the erosion process has been occurring for thousands of years. This action results in a slow retreat of the waterfall upstream.

Because the erosion at Niagara Falls occurs consistently over time, we can assume the rate for the past years will continue into the future given no significant changes happen in the flow or composition of the water. By knowing how much distance has been eroded over a specific timeframe, we can extend this same calculation to predict future erosion distances.

Equation Solving

When faced with a problem requiring prediction, like how far something will erode within a specific time, setting up proportional equations is crucial. Proportions allow us to compare two situations that may otherwise seem different.

In mathematical terms, proportions are relations that state two ratios are equal, and we use them when solving equation problems involving constant rates, as with the Niagara Falls erosion. \[\frac{9.8}{10,000} = \frac{D}{22,000}\]This equation correctly sets two time-related ratios equal, helping us learn the future distance \(D\) the falls will erode. Solving this requires a simple cross-multiplication technique, reflecting how relationships stay balanced.

Mathematical Reasoning

Mathematical reasoning reflects the logical thought processes used to develop solutions from data and known laws or formulas.

• Start by understanding the problem statement: how much erosion occurred, over what time, and what prediction is needed. • Recognize that if conditions remain consistent, historical data provides strong predictive power.

By thinking through all these facets, you see why proportion equations work here. This reasoning not only aids in this task but also supports learning and solving broader mathematical problems. The logic used to tackle the erosion prediction can transfer to many other rate-related endeavors in science and engineering, cultivating problem-solving skills comprehensively.