Thank you,

Cindy Lin

]]>Be prepared. The explanation is hard to visualize.

If the earth were stationary with regard to the sun, the moon would pass between the earth and sun 13 times per year. But because the earth orbits the sun in the same direction as the moon orbits the earth, the moon passes between the earth and sun only 12 times per year. (Compare this to the ending of Around the World in 80 Days. Same math.)

]]>The answer key says to take the ratio of the approximations. It is much better to take the ratio of the formulas. The ratio of the formulas quickly reduces to 4^3, which is 64. This requires much less computation than the ratio of the approximations.

]]>I have always advocated deriving an exact answer first, leaving any approximations until the very last step.

The exact answer works out to 84,000,000,000π cubic miles.

Approximating π as 3.14 (3 figures) gives a final approximation of 263,760,000,000 cubic miles. Compare this to the published answer of 263,810,000,000 cubic miles.

84 is a multiple of 7, so approximating π as 22/7 gives a nicely round final approximation of 264,000,000,000 cubic miles.

Using Excel’s 19-digit approximation of π and rounding the answer to 5 figures gives an answer of 263,890,000,000 cubic miles. Approximating π as 355/113 gives the same 5-figure answer.

]]>Do we have an update on the larger wall art yet? ]]>

~ Mathnasium Matters ]]>