Problem 102 - Applied Calculus

Question

GENERAL: Waterfalls 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 221.44 mph.

Step 1: Understand the Given Formula

The formula given for the speed of water falling from a waterfall is \( \frac{60}{11} x^{0.5} \), where \( x \) is the height of the waterfall in feet. This formula calculates the speed in miles per hour.

Step 2: Identify the Known Values

We know that the height of Ribbon Falls, the highest waterfall in the United States, is 1650 feet. We will substitute this value into the given formula.

Step 3: Substitute the Height into the Formula

Substitute \( 1650 \) into the formula for \( x \): \[ \text{Speed} = \frac{60}{11} \times 1650^{0.5} \]

Step 4: Calculate the Square Root

To simplify the expression, calculate the square root of 1650. \[ 1650^{0.5} = \sqrt{1650} \approx 40.62 \]

Step 5: Compute the Final Speed

Now, multiply the result from Step 4 by \( \frac{60}{11} \): \[ \text{Speed} \approx \frac{60}{11} \times 40.62 \] Calculate the multiplication:\[ \frac{60}{11} \approx 5.4545 \]\[ \text{Speed} \approx 5.4545 \times 40.62 \approx 221.44 \text{ mph} \]

Step 6: Provide the Result

The speed of water at the bottom of Ribbon Falls is approximately 221.44 miles per hour.

Key Concepts

Waterfall Calculations

Numerical problems involving waterfalls often require calculations related to the speed or velocity of the water as it falls. The formula used here, \(\text{Speed} = \frac{60}{11} x^{0.5}\), directly relates the height of the waterfall to the speed of the water hitting the ground. This type of calculation is important in applied calculus because it helps visualize physical scenarios with mathematical precision. Let's break down the given formula:- **Numerator (60):** This number helps convert the speed into miles per hour.- **Denominator (11):** Balances the formula and adjusts the speed calculation.- **Exponent (0.5):** Indicates the square root of the height, effectively reducing large numbers into manageable calculations.The applied formula provides a straightforward means to connect the height of a waterfall with the final speed of the falling water, demonstrating how factors like height directly influence velocity.

Square Root Computations

Square root computations are a cornerstone of this exercise. When the formula includes \((x^{0.5})\), it means we need to compute the square root of the height in feet. This step simplifies numbers for easier calculation.Finding the square root can be done using a calculator or estimating by hand. Here's what you should know:- **Calculators:** Most scientific calculators can easily compute square roots by pressing the "\(\sqrt{}\)" button followed by entering the number.- **Estimation:** If a calculator isn't handy, estimate the square root by finding the nearest perfect squares. For example, since 1650 is between 1600 and 1700, and the square roots of these are 40 and 41, it helps guide the estimation process.- **Approximation Benefits:** Results in more practical and manageable numbers for subsequent calculations.Using these methods makes obtaining a precise square root quicker, fostering a better understanding of numerical relationships in physics and engineering.

Velocity Formulas

Velocity formulas, such as \(\frac{60}{11} x^{0.5}\), play a pivotal role in determining the final speed of objects in motion due to gravitational pull. These formulas use basic principles from physics, incorporating variables to find practical applications in real life.- **COU Connection:** Gravitational pull is a natural force that attracts any two masses towards each other, impacting velocities of anything falling.- **Formula Structure:** Key elements include constants, such as \(\frac{60}{11}\), and variables, such as the height \(x\), which directly influence the speed outcome.- **Units:** The formula outputs in miles per hour, making it relevant to everyday contexts where those measurements are commonplace.Understanding these formulas helps predict how fast objects, like water from a waterfall, travel after falling certain distances. They allow for quick knowledge transfer between theoretical physics and practical, observable behavior in nature.