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Red Ribbon Around the Earth

by Andrew
(New York)











































A red ribbon is tied tightly around the entire earth at the equator. How much more ribbon would you need if you wanted to raise the ribbon exactly one foot above the equator (i.e. pulling the ribbon exactly one foot away from the earth, NOT one foot towards the north pole) everywhere?
What would the answer be if you used a basketball instead of the earth?
(Show or explain your procedure along with your answer.)

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Oct 17, 2011
Red Ribbon Around the Earth
by: Staff


Question:

by Andrew
(New York)

A red ribbon is tied tightly around the entire earth at the equator. How much more ribbon would you need if you wanted to raise the ribbon exactly one foot above the equator (i.e. pulling the ribbon exactly one foot away from the earth, NOT one foot towards the north pole) everywhere?
What would the answer be if you used a basketball instead of the earth?
(Show or explain your procedure along with your answer.)


Answer:

The red ribbon is tied in a circle. The length of the red ribbon used is the distance around the circle of red ribbon (the circumference of the circle).

The extra ribbon needed to circle the earth 1 foot above the surface is the difference between the circumference of the larger circle (1 foot above the surface of the earth) and the circumference of the smaller circle (the distance around the earth on the surface).


The formula for the circumference of a circle = 2πr

The units of the radius (and circumference) can be expressed in any units: km, miles, feet, etc.


1st (smaller) Circumference = length of the ribbon tied tightly around the entire earth at the equator.

2nd (larger) Circumference = length of the ribbon which circles the earth at the equator exactly 1 foot above the surface of the earth.


Extra Ribbon = 2nd (larger) Circumference - 1st (smaller) Circumference

r₁ = 1st (smaller) radius in feet

r₂ = 2nd (larger) radius

Extra Ribbon = 2πr₂ - 2πr₁

r₂ = r₁ + 1 (r₂ is exactly 1 foot larger than r₁)

Substitute r₁ + 1 for r₂

Extra Ribbon = 2π(r₁ + 1) - 2πr₁


Eliminate the parentheses using the distributive law

Extra Ribbon = 2π * r₁ + 2π * 1 - 2πr₁

Extra Ribbon = 2πr₁ + 2π - 2πr₁


Combine like terms

Extra Ribbon = 2πr₁ - 2πr₁ + 2π

Extra Ribbon = 0 + 2π

Extra Ribbon = 2π feet

Extra Ribbon = 6.2831853071796 feet

The final answer is:

the extra ribbon needed to raise the ribbon exactly one foot above the equator = 2π feet, or ≈ 6.2831853071796 feet


Note that the UNITS DO MATTER.

If you raise the ribbon 1 meter above the surface of the earth, the extra ribbon needed would be 2π meters. If you raise the ribbon 1 mile above the surface of the earth, the extra ribbon needed would be 2π miles.


What would the answer be if you used a basketball instead of the earth?

The answer will always be 2π units.

If the basketball has a radius of 5 inches and the ribbon circles the basketball 1 inch above the service, the extra ribbon needed would be 2π inches.

If the ribbon circles the basketball exactly 1 foot above the surface of the basket ball, the extra ribbon needed will be 2π feet. Note that this is exactly the amount of extra ribbon needed to raise the ribbon above the surface of the earth.



Thanks for writing.

Staff
www.solving-math-problems.com


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