Problem 59 - Calculus Early Transcendentals: Pearson New International Edition

Question

Estimate the length of the equator in feet. Assume the radius of the earth to be 4000 miles.

Step-by-Step Solution

Answer

The estimated length of the equator is approximately 132,710,400 feet.

Step 1: Understand the Formula

To find the length of the equator, or the circumference of Earth, we use the formula for the circumference of a circle: \( C = 2\pi r \), where \( C \) is the circumference and \( r \) is the radius. In this case, \( r = 4000 \) miles.

Step 2: Calculate Circumference in Miles

Substitute \( r = 4000 \) miles into the circumference formula: \( C = 2 \pi \times 4000 \). This gives us \( C = 8000\pi \) miles.

Step 3: Convert Miles to Feet

Since there are 5280 feet in a mile, we convert miles to feet by multiplying the circumference in miles by 5280. So, \( C = 8000\pi \times 5280 \) feet.

Step 4: Estimate Using \( \pi \)

Estimate \( \pi \approx 3.14 \). Then \( C \approx 8000 \times 3.14 \times 5280 \). Performing the multiplication, \( 8000 \times 3.14 = 25120 \), and then, \( 25120 \times 5280 \) gives us \( C \approx 132,710,400 \) feet.

Key Concepts

Radius of the Earth

The radius of the Earth is a crucial measurement when calculating the circumference, which is essentially the boundary length of a circle. Picture this as the Earth is a perfect sphere, though in reality, it's slightly flattened at the poles. The average radius is about 4,000 miles, which has been determined through various methods over time.

When used in calculations, the radius helps in determining key geographical and astronomical dimensions.

• The Earth's approximate radius simplifies complex calculations in various fields, such as astronomy and geophysics.
• This estimation is vital because it provides an average measurement useful for all general purposes.

Using the radius, we can derive other measurements, like the circumference, using the formula for the circumference of a circle.

Conversion of Units

Unit conversion is a handy tool in mathematics where one measures quantity in different units. To estimate the circumference of the Earth in feet, you need to convert miles to feet. In this context, knowing that one mile equals 5280 feet is essential.

Here’s how it works in our calculation:

• We calculate the circumference in miles first using the formula: \( C = 2\pi r \).
• Since the circle's radius is in miles, our initial result (8000\( \pi \) miles) gives us the circumference in miles.
• To convert to feet, multiply by the conversion factor, 5280 feet per mile.

This multiplication allows precision and real-world application, ensuring that the final answer spans the context needed, such as architecture or land surveying.

Mathematical Estimation

Mathematical estimation simplifies complex numbers and is an important skill. Estimations allow us to reach approximate answers quickly once we understand the parameters involved. It is helpful when exact numbers are not needed, or preliminary calculations are required.

In our exercise, estimating \( \pi \approx 3.14 \) makes calculations more straightforward.

• The number \( \pi \) is an irrational number, so using 3.14 makes it easier to compute multiplication without a calculator.
• Estimating the final product helps check our result logically, ensuring we understand each step in solving the equation.

Using estimation gives us a ballpark figure, which, in practice, is often sufficiently accurate for many real-world applications, especially when dealing with large measurements like Earth’s circumference.