Frosty Is Melting!

Age 14 to 16
Challenge Level

 

Frosty the snowman is made from two uniform spherical snowballs, of initial radii $2R$ and $3R$. The smaller (which is his head) stands on top of the larger.
 

As each snowball melts, the radius of each snowball decreases by the same amount each minute. During melting each snowball remains spherical and uniform.
 

  • When Frosty is half his initial height, show that the ratio of his volume to his initial volume is 37 : 224 .
     
  • What is this ratio when Frosty is one-tenth of his initial height?

 

Adapted from STEP Mathematics I, 1991, Q2. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.