Suppose that the surface of the earth is smooth and spherical and that the distance around the equator is 25,5

Question

Suppose that the surface of the earth is smooth and spherical and that the distance around the equator is 25,500 miles. A steel band is made to fit tightly around the earth at the equator, then the band is cut and a piece of band 21 feet long is inserted. To the nearest inch, what will be the gap, all the way around, between the band and the earth's surface? (Use 3.14 as an approximate value of Pi)

Best Answer is given to the first correct answer AND who shows the best work.

Answer 1

its gonna work

there are OCEANS around the equator

that tape would sink

a cool question though

if you use that value of pi , the 21 feet won't matter, at least 3.14159 and ten more places I think

Answer 2

The key is the perimeter formula for a circle (the earth equator is a circle)
p=Pi*2*r,
where r is the earth's radius. Setting Pi=3.14:
p=3.14*2*r
Simplifying:
p = 6.28*r

Since the perimeter is 25500, and we need a final result in inches, we'll convert it multiplying by 5280 and then by 12 (1 mile =5 280 feet and 1 feet = 12 inches)
So 25500 miles= 25500x 5280 x 12=1615680000 inches

Now we get earth's radius:
1615680000= 6.28*r
r=p/6.28=1615680000/6.28=257273885.35031846 inches. (ra)

If we add to the 25500 miles (1615680000 inches) 21 feet (that's 21x12=252 inches), we get 1615680252 inches as new perimeter.
With this data, new radius is:
1615680252=6.28*r
r=p/6.28=1615680252/6.28=257273925.477707 inches. (rb)

The difference (rb)-(ra) is the gap:

257273925.477707 -257273885.35031846 = 40.127388536930084, rounded to 40 inches.

Notice that we could have gotten this result in an easier way:
since we are calculating only the difference between radius a (ra) and radius b (rb)
diff=rb-ra=pb/Pi -pa/Pi = (pb-pa)/Pi,
where pa is the earth's perimeter (distance around equator) and pb is the incremented perimeter. But pb-pa=21 x12=252 inches, so diff= 252/Pi =40.12738853503184! which is the same result we got. So it didn't depend on the diameters, but only on the gap!

Answer 3

It does not matter how big the earth is, if you increase the circumference of a circle by 21 feet, the radius will increase by 21 /2*Pi. If we have to use Pi=3.14, then the radius is increased by 3.3439 feet or 40.12 inches. That will be the gap you will have all around: 40 inches.

Answer 4

Known ratios:
Pi = 3.14159265
1 mile = 5280 feet = 63360 inches
1 foot = 12 inches

Circumference = 25500 miles = 134640000 feet
Radius = (Circumference) / (2 * pi) = 4058.45 miles = 21428639.6 feet = 257143675.6 inches

New circumference = 25500 miles + 21 feet = 134640021 feet
New radius = (new circumference) / (2 * pi)
= 21428642.98 feet = 257143715.76 inches

Gap = (new radius) - (old radius)
= 257143715.76 - 257143675.6 = 40.161 inches

Answer 5

Circumference = 25000 mi. = 134640000 ft. = 2827440000 in.
Diameter = (2827440000 / 3.14) = 900458598.7 in.
Radius = 450229299.4 in

Add 21 inches to circumference to get: 2827440021 in.
New diameter = (2827440021 / 3.14) = 900458605.4 in
New radius = 450229302.7 in
New radius exceeds old radius by 3 inches.

Answer 6

After doing the calculations I have concluded that the gap would be so narrow that it would be too insignificant to measure.

Answer 7

yeah but the surface of the earth isn't like that so what's the point of this question it's superflous