Fun math story problem for MLK Day?

Question

Show all work, please!

(1) Suppose the earth were a perfectly smooth sphere with a radius of 4,000 miles, and suppose there were a solid steel band running around the equator. You want to raise this band off the ground exactly 1 inch so that everywhere around the equator the band is exactly 1 inch off the ground. How much longer is the band now? In other words, if the original length of the steel band was "A", and the new length is "B" what is B - A?

(2) Suppose you have a basketball which is a perfectly smooth sphere with a radius of 12 inches, and suppose there were a piece of string running around the "equator" of the basketball. You want to raise this piece of string off the surface of the basketball exactly 1 inch so that everywhere around the basketball the string is exactly 1 inch off the surface. How much longer is the string now? In other words, if the original length of the string was "D", and the new length is "E" what is E - D?

Which answer is bigger (1) or (2)?

Answer 1

C = 2*pi*r
dC/dr = 2*pi
dC=2*pi*dr
So if we change the radius by 1 inch, then C changes by 2*pi inches. The same is true of the basketball. It makes no difference what size the sphere is.

A change of dr in the radius causes a change of dC in the circumference regardless of size of radius.

Answer 2

This is my thinking for (1):

The radius of the earth in inches is:

4000 mi * 5280 ft/mi * 12 in/ft = 253440000 in

So the circumference at the equator is 2*pi*r or 6.28*253440000 =

1591603200 inches

If you add 1in to the radius to lift the band 1 inch off the equator:

4001 * 5280 * 12 = 253503360 in

So the new circum is 2 * pi * 253503360 = 1592001101

The band needs to be 397901 in longer, or 6.28 mi longer

Is that right?

Answer 3

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Answer 4

Your first answer is the best (in my opinion), tho more abstract, it is the most elegant.

You can look at it this way too:

r1 = 2π(r1)
r2 = 2π(r2)

r2 - r1 = 2π(r2) - 2π(r1) = 2π(r2 - r1)

If the difference is the same (r2 - r1), then it is simply a constant multiplied to 2π......regardless of the size of the original radius.

Answer 5

it will be

1.2pi[(4000*5280+1)-4000*5280]
=2pi

2.2pi[(12+1)-12]
=2pi