Problem 93 - 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 world, Angel Falls in Venezuela (3281 feet high).

Step-by-Step Solution

Answer

The speed of the water at the bottom of Angel Falls is approximately 312.64 miles per hour.

Step 1: Identify Given Values

We are given the formula for the speed of water falling from a height \(x\) feet: \(\frac{60}{11} x^{0.5}\) miles per hour. We also know the height of Angel Falls is 3281 feet. Our task is to find the speed of the water at this height.

Step 2: Substitute the Known Value

Substitute \(x = 3281\) into the speed formula. The expression becomes \(\frac{60}{11} \times (3281)^{0.5}\).

Step 3: Calculate the Square Root

Calculate the square root of 3281. \((3281)^{0.5} \approx 57.28\).

Step 4: Calculate the Speed

Using the result from Step 3, substitute into the expression: \(\frac{60}{11} \times 57.28\). This results in: \(\approx 312.64\) miles per hour.

Key Concepts

Speed Calculation

In applied calculus, speed calculation involves the use of mathematical formulas to determine how fast something is moving. In our given problem, we are working with a specific formula to calculate the speed of water falling from a height. The formula provided is \( \frac{60}{11} x^{0.5} \) miles per hour, where \( x \) represents the height in feet.

• This formula indicates that the speed of the water increases as the height increases.
• The \( x^{0.5} \) term signifies taking the square root of the height \( x \).
• The constant \( \frac{60}{11} \) scales the speed after applying the square root.

By substituting the height of Angel Falls, which is 3281 feet, into this formula, we begin the process of calculating the speed at which the water hits the ground. It requires calculating the square root of 3281 and then multiplying the result by \( \frac{60}{11} \), ultimately giving us the speed in miles per hour. Understanding this formula helps in applying it accurately to different scenarios involving falling objects.

Waterfall

Waterfalls provide an intriguing natural context for applied calculus. At Angel Falls, the world's highest waterfall, water descends a staggering 3281 feet. This drop provides a perfect real-world example of how physics and calculus can explain natural phenomena like a waterfall's speed.

• The height of a waterfall is crucial because it determines the potential energy available before the fall.
• As the water falls, its potential energy is converted into kinetic energy, contributing to its speed.
• This transformation is governed largely by the force of gravity, simplified in our problem by ignoring air resistance.

The extreme height of Angel Falls makes it an excellent subject for studying the physics of falling objects. With calculus, we can precisely quantify aspects of the fall, such as the speed with which the water hits the ground. In this case, understanding the waterfall's height and dynamics becomes essential for accurate speed calculations.

Physics Application

This exercise illustrates a fundamental physics application that combines gravitational forces with calculus equations. The underlying principle is that objects falling freely under gravity, like water in a waterfall, accelerate at a rate determined by the force of gravity.

• As water falls, gravitational acceleration increases its speed, proportional to the height of the fall.
• The formula \( \frac{60}{11} x^{0.5} \) encapsulates these physics principles into a workable equation suitable for speed calculation.
• By excluding variables like air resistance, the problem focuses purely on the gravitational aspect.

The example of Angel Falls shows how theoretical physics can be applied practically. Physics applications like these allow us to predict behavior, plan safety measures, and even harness natural phenomena for human benefit. Calculus becomes a valuable tool in transforming the physical world into understandable and measurable terms.