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Make Me A Valentine, She Said

Two boys, Peter and Steven, are interested in taking Linda to the valentine dance. Linda, however, wants to make sure that any boy she dates is a sharp math student, so she tells each of them that he must make her a valentine heart. In addition, he must tell her the area of his heart, and the equation that would trace out its perimeter. In addition, Linda is picky and won't accept an answer without an explanation.

Peter is in calculus, and with the help of his calculus book comes up with the equation r = 1+sin(theta). He also puts together a very elaborate and correct explanation for Linda.

Steven was looking over Peter's shoulder as Peter worked on his equation and explanation. Steven thinks Linda would like a bigger valentine, so he writes down the equation r = 2+2sin(theta) and figures that since his equation is twice Peter's equation, the area of his heart will be twice as big, too.

What are the areas of Peter's and Steven's valentine hearts? Is Steven's reasoning correct?

 

Source: mathforum.org/librarypow/puzzles/index2.ehtml?puzzle=1482


Hint: Draw the graphs of each of the two curves, use the formula for area in polar coordinates.

 


Solution Commentary: Assuming you have answered the question, discuss the merits of Steven's reasoning. What would have happened if he had used any of the following (i.e. compare the areas):

  • r = .5+.5sin(theta)
  • r = 2sin(theta)
  • r = 1+2sin(theta)
  • r = .5+2sin(theta)
  • r = 2+sin(theta)
Is there a generalization that can be formulated and shared with Steven? For example, suppose you are given r=a+bsin(theta) and r=c+dsin(theta). Can they ever have the same area without both a=c and b=d? If yes, can you establish the necessary conditions?