Problem 94 - Brief Applied Calculus

Question

Water falling from a waterfall that is \(x\) feet high will hit the ground with speed \(\frac{60}{11} x^{0.5}\) miles per hour (neglecting air resistance). Find the speed of the water at the bottom of the highest waterfall in the United States, Ribbon Falls in Yosemite, California (1650 feet high).

Step-by-Step Solution

Answer

The speed is approximately 221.56 mph.

Step 1: Identify Given Information

We are provided the height of the Ribbon Falls, which is 1650 feet. The formula for the speed of water hitting the ground is given by:\[ v = \frac{60}{11} x^{0.5} \]where \( x \) is the height of the waterfall in feet.

Step 2: Substitute Height into Speed Formula

Substitute the given height of the waterfall (1650 feet) into the speed formula:\[ v = \frac{60}{11} (1650)^{0.5} \]

Step 3: Calculate the Square Root

Compute the square root of 1650:\[ (1650)^{0.5} \approx 40.62 \]

Step 4: Substitute and Simplify

Substitute the square root value back into the formula and simplify:\[ v = \frac{60}{11} \times 40.62 \]

Step 5: Calculate the Speed

Perform the multiplication and division:\[ v \approx \frac{60 \times 40.62}{11} \]\[ v \approx \frac{2437.2}{11} \]

Step 6: Final Result

Evaluate the division to find the speed:\[ v \approx 221.56 \]This means the speed of the water at the bottom of Ribbon Falls is approximately 221.56 miles per hour.

Key Concepts

Waterfall Speed Calculation

Waterfalls provide a beautiful and intriguing real-world example to explore speed calculations using applied calculus principles. Understanding how fast water cascades down a waterfall involves using mathematical formulas to derive precise answers. When dealing with the problem of calculating a waterfall's speed, we start by using a specific formula to understand the nature of the forces involved.

In our exercise, the speed of water hitting the ground is calculated through the formula \( v = \frac{60}{11} x^{0.5} \). Here, the variable \( x \) represents the waterfall's height in feet. By substituting the height of the waterfall, we can easily determine the speed at which the water impacts the ground. This specific formula provides a direct approach to solving for speed and showcases an example of how mathematical equations are utilized to solve real-world problems.

Real-World Calculus Applications

Calculus is often viewed as an abstract branch of mathematics; however, its applications in the real world are vast and impactful. Calculating the speed of water from a waterfall is one such example. This type of problem demonstrates how calculus not only helps in theoretical mathematical computation but also in understanding natural phenomena.

When applying calculus to waterfall speed, we employ the concept of instantaneous rates to analyze moving objects. The formula provided uses functions and derivatives to convey how height influences speed. This is analogous to understanding how different factors may affect rates of change in various fields such as physics, engineering, and economics.

• It bridges the gap between theoretical and practical understanding.
• Helps in analyzing real-world movement and forces at play.
• Demonstrates the utility and necessity of calculus in scientific inquiries.

Mathematical Modeling

Mathematical modeling is a powerful tool that assists scientists and engineers in converting real-world situations into manageable mathematical expressions. In the case of the waterfall speed exercise, we use a mathematical model to represent the complex physical processes occurring as water falls.

This model simplifies the problem by allowing us to work with achievable calculations and reasonable assumptions, such as neglecting air resistance. With this approach, we can predict and comprehend essential elements such as the speed at which water will hit the ground when falling from a specific height. This kind of modeling is vital in many fields as it turns intricate scenarios into solvable problems using mathematical functions and equations.

• It offers a simplified understanding of natural events.
• Enhances predictive analysis capabilities.
• Supports innovations across various scientific and commercial sectors.