Problem 10 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

When designing the one-third-of-a-mile-long Georgia World Congress Center, the building that housed nearly one-fifth of the events of the 1996 Olympics, engineers had to take into account the curvature of the earth (Sports Illustrated, August 5,1996 ). Assuming a constant curvature of the earth, how many feet would it curve in one-third of a mile? In other words, assume a cross-section of the earth is a perfect circle and draw a tangent line to the curve of this circle at one end of the building. How far away would the tangent line be from the circle itself at the other end of the building? (Use 3960 miles as the radius of the earth.)

Step-by-Step Solution

Answer

The tangent line would be approximately 1 foot 5 inches away from the circle itself at the other end of the building.

Step 1: Converting units

First, convert the length of the building from miles to feet and the radius of the Earth from miles to feet because the curvature must be found in feet. Knowing that 1 mile is equivalent to 5280 feet, the length of the building becomes \(\frac{1}{3} * 5280\) feet and the radius of the Earth is \(3960 * 5280\) feet.

Step 2: Applying Small-Angle Approximation

In this step, the small angle approximation is used, assuming that 's' represents the arc length, 'r' the radius, and 'θ' the angle in radians, we have \(s = r * θ\). Here, the length of the building is considered as the arc length. So, it is possible to isolate 'θ' in the equation resulting in: \(θ = \frac{s}{r} = \frac{\frac{1}{3} * 5280}{3960 * 5280}\).

Step 3: Calculating Curvature

The curvature is calculated as the distance from the tangent line to the circle. This can be achieved by subtracting the radius of the Earth from the distance of the tangent line from the center of the Earth. In this scenario, the distance of the tangent line from the center of the Earth is given by \(r * cos(θ)\). Hence, the distance away would be \(r - r * cos(θ)\). Substituting the value of 'r' and 'θ' from above to find the answer.

Key Concepts

Tangent Line

In geometry, a tangent line is a straight line that touches a curved surface or circle at exactly one point. This touch point is known as the point of tangency. The tangent line does not cross the curve at this point.
To visualize it, think of a circle with a single line just kissing its surface. The significance of this in our context with the Earth's curvature is that the tangent line represents the shortest possible distance over a stretch, like the length of the building in the exercise.

• The tangent line is perpendicular to the radius at the point of tangency.
• It helps in visualizing how much the Earth's surface curves away over a distance.
• In the calculation of the Earth's curvature, it allows us to determine how far off the surface the tangent line will be at the opposite end of the building.

By understanding the role of the tangent line, we can calculate the deviation caused by Earth's curvature, which is crucial for precise engineering in large constructions.

Small-Angle Approximation

The Small-Angle Approximation is a useful mathematical tool that simplifies calculations involving very small angles. This approximation assumes that for small angles, measured in radians, the sine of the angle, the angle itself, and the tangent of the angle are almost identical.
In our exercise, when the building's length is considered an arc of a circle (representing Earth), we use this approximation to calculate the angle subtended by this arc at the Earth's center. The formula is quite simple:

• \[\theta \approx \sin(\theta) \approx \tan(\theta)\]
• This is used when \(\theta\) is in radians and is a small number.

This simplification is essential for easy calculations, as it allows us to find \(\theta\) using \(s = r \cdot \theta\), where \(s\) is the arc length (distance of the building) and \(r\) is the Earth's radius.
Without the small-angle approximation, handling these calculations would be more complex, especially when dealing with Earth's large radius and relatively short building stretches.

Arc Length

Arc length is a critical concept in geometry that deals with measuring the distance along a curved line, which in this case refers to the segment of the Earth's circumference in contact with the building.
In the context of the exercise, the arc length is the actual length of the path the building traces over the Earth's curved surface:

• The formula for arc length is \(s = r \cdot \theta\), with \(r\) being the radius of the circle and \(\theta\) the central angle in radians.
• This allows us to equate and solve for \(\theta\) using the known values of arc length (building length) and Earth's radius.

Understanding arc length helps in comprehending how distances are measured over curved surfaces, which is vital in many real-world applications. This insight not only aids engineers but also provides valuable knowledge in various scientific fields. It shows how curvature can impact structural design and is crucial to maintain precision over large distances.