Question

Imagine a rope wrapped around Earth at the equator (Earth's circumference \(C\) ). Then think of adding \(d\) feet to the rope's length so it can now circle Earth at a distance \(h\) feet above the equator at all points. Write an equation to model this situation.

Short answer

The equation that models adding \(d\) feet to the circumference in order to raise the rope \(h\) feet above the equator is \(d = 2\pi h\).

Step-by-step solution

Understand the Context

In this case, a rope that was initially wrapped around Earth now elevated to be \(h\) feet above the Equator, due to an additional length of \(d\) feet.

Identify the Original Circumference Equation

The formula used to calculate the original circumference \(C\) is \(C = 2\pi r\), where \(r\) is the radius of Earth.

Calculate the New Radius

By adding the foot of height \(h\) to the radius \(r\) of Earth, we get the new radius \(r'= r + h\).

Calculate the New Circumference Using the New Radius

The new circumference, \(C'\) can be found using the formula \(C' = 2\pi r'\). Substituting \(r' = r + h\) obtained in the last step, we find \(C' = 2\pi (r + h) = C + 2\pi h\). Recall that \(C' = C + d\), setting these equal to each other and simplifying gives the equation \(d = 2\pi h\).

Key concepts

Circumference of Earth

Understanding the circumference of Earth is pivotal to a breadth of disciplines, from geography to aerospace engineering. The circumference is the distance around the Earth along the equator, a great circle. The Earth's shape is nearly spherical, and mathematically, the circumference (\(C\)) of a sphere can be calculated using the formula \(C = 2\pi r\), where \(\pi\) is a constant (approximately 3.14159) and \(r\) is the radius of the sphere.

To envisage this concept, imagine wrapping a rope tightly around Earth's equator; this rope's length equals Earth's circumference. Known measurements estimate the Earth's circumference to be about 24,901 miles or 40,075 kilometers. With this information, scientists, geographers, and navigators can calculate distances around Earth and coordinate GPS technologies for accurate navigation.

Radius Calculation

Radius calculation is a fundamental part of geometry and is intricately linked to understanding circular shapes like spheres. For an object with a spherical shape like Earth, calculating the radius is typically done when the circumference is known using the rearranged formula \(r = \frac{C}{2\pi}\).

In the context of Earth and the exercise provided, if we know Earth’s circumference, we can solve for Earth’s radius. Such calculations are not just academic exercises; they're central to fields like geodesy, which deals with the measurement and understanding of Earth's geometric shape, orientation in space, and gravity field, as well as in various engineering disciplines.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world scenarios through mathematical formulas and equations. It allows us to predict, analyze, and represent complex systems in a simplified manner. The exercise in our context involves generating a mathematical model to represent the situation where a rope, originally fitting the Earth's circumference, is then elevated by a certain height above the equator.

To create this model, you first consider the initial conditions of the Earth and the rope, then describe how these conditions change with certain variables, such as the height (\(h\) feet) added to the Earth's radius. This mathematical representation is not only useful in theoretical studies but also in practical applications such as construction, environmental predictions, economics, and more.

Linear Equations

Linear equations form the cornerstone of algebra and represent relationships where each term is either a constant or the product of a constant and a single variable. They graph as straight lines and have wide-ranging applications in various fields of study.

In the context of the exercise, the relationship between the added height (\(h\) feet) and the additional length (\(d\) feet) of rope needed to maintain a circle around the Earth is a linear one, described by the equation \(d = 2\pi h\). This is a direct proportionality where the constant of proportionality is \(2\pi\), correlating linearly to the concept of the circumference of a circle. Knowing how to work with linear equations enables us to solve for unknown variables and to understand the relationship between quantities in both simple and complex systems.