Question
Assuming that Earth is a perfect sphere using 27,000miles as the Earth's circumference, suppose you wrap a rope around earth so it is tightly pressed against the earth. Next add 10 feet of rope to the rope that is wrapped around earth and spread out the rope so it is the same distance away from the earth all the way around. How far away from earth is the rope now? Write your answer in feet rounded to two decimal places.
Answers
GPT-4o mini
To solve this problem, we begin by determining the radius of the Earth based on the given circumference.
The formula for the circumference \( C \) of a circle is given by:
\[
C = 2\pi r
\]
where \( r \) is the radius. Given that the Earth's circumference is 27,000 miles, we can find the radius as follows:
\[
r = \frac{C}{2\pi} = \frac{27000 \text{ miles}}{2\pi}
\]
Now we convert the miles to feet since we will need the distance in feet. There are 5280 feet in a mile:
\[
27000 \text{ miles} = 27000 \times 5280 \text{ feet} = 142560000 \text{ feet}
\]
Substituting this value into the radius formula, we have:
\[
r = \frac{142560000}{2\pi} \approx \frac{142560000}{6.28318} \approx 22714637.211 \text{ feet}
\]
Next, we add 10 feet to the original length of the rope:
\[
L = 142560000 \text{ feet} + 10 \text{ feet} = 142560010 \text{ feet}
\]
Now, we need to find the new radius \( R \) of the circle formed by this new length of rope:
\[
L = 2\pi R
\]
Solving for \( R \):
\[
R = \frac{L}{2\pi} = \frac{142560010}{2\pi} \approx \frac{142560010}{6.28318} \approx 22714637.30 \text{ feet}
\]
The distance from the Earth's surface to the rope is the difference between the new radius \( R \) and the original radius \( r \):
\[
\text{Distance} = R - r = 22714637.30 - 22714637.211 = 0.089 \text{ feet}
\]
Therefore, the rope is approximately \( 0.09 \) feet away from the surface of the Earth, when rounded to two decimal places.
\[
\boxed{0.09 \text{ feet}}
\]
The formula for the circumference \( C \) of a circle is given by:
\[
C = 2\pi r
\]
where \( r \) is the radius. Given that the Earth's circumference is 27,000 miles, we can find the radius as follows:
\[
r = \frac{C}{2\pi} = \frac{27000 \text{ miles}}{2\pi}
\]
Now we convert the miles to feet since we will need the distance in feet. There are 5280 feet in a mile:
\[
27000 \text{ miles} = 27000 \times 5280 \text{ feet} = 142560000 \text{ feet}
\]
Substituting this value into the radius formula, we have:
\[
r = \frac{142560000}{2\pi} \approx \frac{142560000}{6.28318} \approx 22714637.211 \text{ feet}
\]
Next, we add 10 feet to the original length of the rope:
\[
L = 142560000 \text{ feet} + 10 \text{ feet} = 142560010 \text{ feet}
\]
Now, we need to find the new radius \( R \) of the circle formed by this new length of rope:
\[
L = 2\pi R
\]
Solving for \( R \):
\[
R = \frac{L}{2\pi} = \frac{142560010}{2\pi} \approx \frac{142560010}{6.28318} \approx 22714637.30 \text{ feet}
\]
The distance from the Earth's surface to the rope is the difference between the new radius \( R \) and the original radius \( r \):
\[
\text{Distance} = R - r = 22714637.30 - 22714637.211 = 0.089 \text{ feet}
\]
Therefore, the rope is approximately \( 0.09 \) feet away from the surface of the Earth, when rounded to two decimal places.
\[
\boxed{0.09 \text{ feet}}
\]