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A New Sport: Melt Racing

Oh no! It is 8:30 pm on Christmas Eve and Frosty has been locked in the greenhouse by the evil magician again. He must get out before he melts into water.

Frosty is made out of two spherical balls of snow such that one is sitting on top of the other (typical snowman for most snowmen!). The upper ball of snow has diameter 18 inches and the lower one has diameter 36 inches, both prior to melting. Frosty is made of Christmas snow which melts at a different rate than normal snow. Each ball will melt at a rate of 125 in3 per minute while in the greenhouse.

  • At what time will the upper ball melt?
  • At what time will the lower ball melt?
  • At what time will Frosty be made up of exactly half the amount of snow as when he began his greenhouse adventure?
  • What is the rate of change of the radius of the lower ball at that moment?
  • If Santa could use his magic to adjust the rate that Frosty melts so that part of him is still left by Christmas (so that he could be rescued by Santa), what would the new rate be?


Source: POW offered on a now-defunct AP Calculus web-site

Hint: Even if you do not know claculus, you can attack this problem.

  • Set up a table (with time increments) to show the changing size of the snowballs
  • Generalize this table in the form of two equations, which can be graphed...analyzed, etc.


Solution Commentary: Sorry, remember the website that offered this problem was defunct...and the problem's solution has melted away into internet deadspace as well!