A person 6 feet tall walks around the earth. How many feet farther does the top of this person's head travel than the bottom of his feet? Express your answer in terms of Pi.
circumference = 2 * π * radius
if the radius is increased by 6 ft , how much does the circumference increase?
To find the number of feet farther the top of the person's head travels than the bottom of his feet as he walks around the earth, we need to calculate the difference in circumference between two circles.
The first circle represents the path traced by the bottom of his feet while walking, and its circumference can be calculated using the formula:
C1 = 2πr1
Here, r1 represents the radius of the earth, which is the distance from the center of the earth to the bottom of his feet.
The second circle represents the path traced by the top of his head while walking, and its circumference can be calculated in a similar manner:
C2 = 2πr2
Here, r2 represents the radius of the earth, which is the distance from the center of the earth to the top of his head.
Since the person's height is 6 feet, the difference in radius between the top of his head and the bottom of his feet would also be 6 feet. Therefore, we can write:
r2 = r1 + 6
Substituting this value into the equation for C2, we have:
C2 = 2π(r1 + 6)
Now, the difference in circumference is C2 - C1:
C2 - C1 = 2π(r1 + 6) - 2πr1
Simplifying this expression gives:
C2 - C1 = 2πr1 + 12π - 2πr1
The two πr1 terms cancel each other out, leaving:
C2 - C1 = 12π
Therefore, the top of the person's head travels an extra distance of 12π feet compared to the bottom of his feet as he walks around the earth.
To answer this question, we first need to find the distance the person travels around the Earth. Let's assume the Earth's circumference is C (we will calculate this value later).
To calculate the distance the person's head travels, we need to imagine a circle around the Earth formed by the head's path. The circumference of this circle will be equal to the distance traveled by the person's head.
Now, to find the circumference of the circle formed by the person's head (C_head), we can use the formula: C_head = 2πr_head.
Given that the person is 6 feet tall, we can assume that the radius of the head's circle (r_head) is 6 feet.
Now, to find the distance the person's feet travel, we need to imagine another circle formed by the feet's path. The circumference of this circle will be equal to the distance traveled by the person's feet.
To find the circumference of the circle formed by the person's feet (C_feet), we can use the same formula: C_feet = 2πr_feet.
Given that the person is 6 feet tall, we can assume that the radius of the feet's circle (r_feet) is also 6 feet.
Now we have the following equations:
C_head = 2πr_head = 2π(6) = 12π.
C_feet = 2πr_feet = 2π(6) = 12π.
The difference in distance traveled between the top of the person's head and the bottom of their feet can be found by subtracting the circumference of the circle formed by the feet from the circumference of the circle formed by the head.
Therefore, the difference in distance traveled is given by:
12π - 12π = 0.
Hence, the top of the person's head does not travel any farther than the bottom of their feet around the Earth.