Question

The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles; one by wrapping a rope around the equator and then a second circle that is \(38 \mathrm{ft}\) longer than the first rope (see figure). How much space is between the ropes?

Short answer

Answer: The approximate distance between the two concentric circles is 0.4 feet.

Step-by-step solution

Determine the radius of the Earth at the equator

We are given the Earth's equatorial circumference, so we can use the formula for the circumference of a circle to find its radius (\(r_1\)): \(C_1 = 2\pi r_1\). Then we'll convert the given circumference from miles to feet. Note that 1 mile = 5280 ft. Circumference at equator, \(C_1 = 24882 \, \text{miles} \times 5280 \, \text{ft/mile} = 131332960 \, \text{ft}\) Now calculate the radius, \(r_1\): \(r_1 = \frac{C_1}{2\pi} = \frac{131332960}{2\pi} \approx 20902084.5 \, \text{ft}\)

Determine the circumference and radius of the second circle

We know that the length of the second rope, which forms the second circle, is 38 ft longer than the first one. Thus, its circumference is \(C_2 = C_1 + 38 \, \text{ft}\). Then, we'll apply the circumference formula and find the radius of the second circle (\(r_2\)). Circumference of second circle, \(C_2 = 131332960 + 38 = 131332998 \, \text{ft}\) Now calculate the radius, \(r_2\): \(r_2 = \frac{C_2}{2\pi} = \frac{131332998}{2\pi} \approx 20902084.9 \, \text{ft}\)

Calculate the space between the ropes

Finally, we'll find the difference in radii to determine the space between the ropes: \(\mathrm{Space} = r_2 - r_1 \approx 20902084.9 - 20902084.5 = 0.4 \, \text{ft}\) Therefore, the space between the two ropes is approximately 0.4 feet.

Key concepts

Circumference and Radius Calculations

Understanding how to calculate the circumference and radius of a circle is essential in solving many geometry problems, especially those involving circles. The circumference is the total distance around a circle. The formula to find the circumference of a circle is given by the equation: \[C = 2\pi r\]where \(C\) is the circumference and \(r\) is the radius. This formula can also be rearranged to find the radius if the circumference is known:\[r = \frac{C}{2\pi}\]

• For example, if you know the Earth's equatorial circumference is 24882 miles and you want to find the radius, convert the circumference to feet first (using unit conversion) and then apply the formula.
• This is an essential skill in mathematics and physics, where circular measurements are common.

By rearranging the formula, you can find the radius given the circumference, which is a commonly needed calculation in practical applications, such as determining the size or scale of objects on large dimensions, like the Earth's equator.

Unit Conversion in Mathematics

Mathematics often requires converting one unit of measure to another, such as miles to feet. This is vital to ensure calculations are performed correctly, especially when measurements need to be precise. Here's how you can do it:

• Identify the conversion factor. For instance, 1 mile is equal to 5280 feet.
• Multiply the quantity in the original units by the conversion factor to get the desired units.

For example, when calculating the Earth's circumference in feet from miles, multiply 24882 miles by 5280 feet per mile, giving a result of 131332960 feet. Unit conversion ensures that your equations and comparisons remain accurate. Without correct unit conversion, your calculations can result in incorrect conclusions, particularly in engineering, physics, and geography, where precision is essential.

Geometry of Circles

The geometry of circles plays an integral role in many fields, from architecture to nature. A circle is defined by its center and its radius, which is the distance from the center to the circle's boundary.Some key properties include:

• The circumference, as discussed, which is the outer boundary of the circle.
• The radius, which helps in defining the size and dimensions of the circle. The diameter, which is twice the radius, gives the widest measurement across the circle.
• A concentric circle is one that shares the same center. In problems like these, knowing the geometry helps visualize and solve issues where changes, like an additional length in rope around the equator, alter dimensions slightly.

Understanding circle geometry allows for solving not only practical problems involving measurements and spaces but also brings clarity to theoretical problems in mathematics and physics, such as calculating the area of a circle using the formula \[A = \pi r^2\]Learning and applying the principles of circle geometry can demystify many complex-looking problems and make them approachable with logical reasoning.