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: \(\boxed{\frac{9.8}{10,000}=\frac{D}{22,000}}\).