Problem 101 - Applied Calculus

Question

101-102. 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 world, Angel Falls in Venezuela ( 3281 feet high).

Step-by-Step Solution

Answer

The speed at the bottom of Angel Falls is approximately 312.49 mph.

Step 1: Identify the Formula

The problem states that the speed of water falling from a height of \(x\) feet is given by the formula \( v = \frac{60}{11} \cdot x^{0.5} \) miles per hour. Here, \( v \) is the speed, and \( x \) is the height of the waterfall.

Step 2: Substitute the Given Height

We know the height of Angel Falls is 3281 feet. Substitute \( x = 3281 \) into the formula: \( v = \frac{60}{11} \cdot (3281)^{0.5} \).

Step 3: Calculate the Square Root

Calculate \((3281)^{0.5}\), which is the square root of 3281. This will be approximately 57.29.

Step 4: Compute the Speed

Now substitute \(57.29\) into the speed formula: \( v = \frac{60}{11} \cdot 57.29 = \frac{60 \times 57.29}{11} \).

Step 5: Perform the Multiplication and Division

Calculate \(60 \times 57.29 = 3437.4\). Then divide by 11: \(\frac{3437.4}{11} \approx 312.49\).

Step 6: State the Result

The speed of the water at the bottom of Angel Falls is approximately 312.49 miles per hour.

Key Concepts

Speed calculation

Speed calculation is a central concept in physics that enables us to understand how fast an object is moving. In practical terms, speed is the rate at which an object covers distance. In the context of waterfalls, this tells us how quickly the water will hit the ground from a certain height. To calculate the speed of an object, you typically need two core pieces of information:

• Distance: The height or length over which the object travels.
• Time: The period it takes to cover that distance.

However, in this exercise with Angel Falls, we used a specialized formula to directly calculate the speed of falling water from a given height without explicitly considering time. The formula \( v = \frac{60}{11} \cdot x^{0.5} \)considers both gravity and distance by using the height (\(x\)) of the waterfall, providing a simplified method of finding speed neglecting air resistance.

Square roots

Square roots are mathematical operations used to find a number which, when multiplied by itself, gives the original number. For example, if \(x = 3281\), then the square root (\(x^{0.5}\)) represents the number that when multiplied by itself equals 3281. In this exercise, the computation of the square root of 3281 is crucial for determining the final speed of the water. Square roots are widely used in various mathematical models and formulas, particularly involving physics and geometry, because they allow us to simplify expressions or solve equations that involve quadratic terms. Calculating square roots can be performed using calculators or estimation techniques when an exact number isn't required.
Taking the square root of large numbers, as shown by figuring out \((3281)^{0.5}\approx 57.29\), helps simplify our main speed formula in the exercise. This precision may sometimes be adequate unless higher accuracy is necessary for specific engineering calculations.

Mathematical modeling

Mathematical modeling is a process that uses mathematical structures, expressions, and equations to represent real-world phenomena. This approach is invaluable in predicting and understanding complex systems and behaviors. In our exercise about the waterfall, the mathematical model provided by the formula \( v = \frac{60}{11} \cdot x^{0.5} \)allows us to estimate the speed of water based on the height of the waterfall, modeling a real-world physical scenario.Mathematical models help simplify complex problems, making them easier to analyze and solve. Some aspects commonly include:

• Variables: Symbols that represent unknown quantities (e.g., height in our formula).
• Constants: Fixed values that define specific properties of the system (e.g., \( \frac{60}{11} \).
• Functional relationships: Expressions showing how different variables relate to one another.

This model neglects air resistance, which is acceptable for estimative purposes. However, comprehensive models can include more variables for precision. Understanding these simplified models can be a stepping stone to mastering more intricate and comprehensive systems.