Homework: Visualizing Large Numbers: Oil and Surface Area
Use the Google Calculator (or regular calculator plus a source for units conversions) to answer the following.

Although it is difficult to imagine such a large number, consider approximately one trillion (1,000,000,000,000) barrels of oil have been consumed since the discovery of "oil" (petroleum literally means, "rock oil").

  1. Using above approximation, this amount is about 42,000,000,000,000 or 42 trillion gallons of oil. (or 42·1012 which is 4.2·1013 )

  2. The total land area on Earth is about 149,000,000 square kilometers. Use Google Calculator to find out: each square kilometer has the same area as about 10,763,910.4 square feet (not square meters). Round your answer in this step, to the nearest 1,000 sq. feet: 10,764,000

  3. Thus the total land area of Earth is about 1,603,836,000,000,000 or 1.603836·1015 square feet (ft2)

  4. Divide this number (of square feet) by the earlier number of gallons of oil burned so far to conclude:
    "We've burned 1 gallon of oil for every ≈38.187 or ≈38.2 ft2 of land" (round to nearest 0.1 ft2)

    Call this number A. Imagine dividing up all the land area on Earth into parcels or portions each having this area. Thus we've burned one gallon of oil for each such portion, if you can imagine dividing up all land on earth -- not just the inhabited parts, but every bit of land in the Sahara desert, every such bit of land in the Amazon, every such bit of land in every city, town, field, hillside, etc, into portions of this area, imagine a gallon of oil consumed and no longer available, for each such portion.

  5. How big does a circle have to be to have area A? Well, A=π·r2. So divide A by π and then take the square root to find r. This radius r equals about 3.487, so ≈ 3.49 or ≈ 3.5 feet (to nearest 0.01 or 0.1)

  6. Thus, if your arm is about 2 feet long, then if you hold a stick in your hand that is ≈ 1.49 (or ≈1.5) feet long, this would sweep a circle of radius "r" (whose center is in your body). We've burned a gallon of oil for every portion of land-mass on Earth whose area is the same as the circle you imagined sweeping out with your arm.

  7. Suppose it's more accurate to say 1.1 trillion barrels of oil have been consumed to date. What would the new value be for A? Is there a shortcut? What would be the new value for r? Shortcut here? (Warning: ratio of two circles' areas does not equal the ratio of their radii! How much more area does a circle of radius 2 have than a circle of radius 1, for example?)
    Area of π·12 vs. area of π·22 so ratio of 4π / π so 4:1 ratio; NOTE: ratio of areas ≠ ratio of radii!
    New A≈ 34.715 (see below) rounds to ≈34.72 or ≈34.7 ft2 and new r ≈ ≈3.324 or ≈ 3.3 ft. (rounding as before)
    Calculation: We have 1.1 times as much oil, but our quotient for A is
              (area (in ft2)) ÷ (amount of oil (in gallons))
              So new quotient is 1.1 times smaller; so: (1/(1.1))·(38.187)≈34.715
              New radius is found by dividing new area by π and taking square root

              Trickier shortcut to find new radius (this only works before we subtract the 2ft of arm length!) is to divide the old radius by (1.1)½     Aside: when total of oil consumed reaches ≈1.35 trillion barrels (which we are not very far from) r≈3 almost exactly, so assuming body-center to hand radius is still 2 feet, making a circle with a 1-foot ruler would show you what size circle is the amount of land area, such that for every piece of dry land on Earth, a gallon of oil have been burned for that (area equivalent) section of Earth.