Canny fraction
What fraction of the volume of this can is filled with lemonade?
Age
14 to 16
Challenge level
Problem
Image
A cylinder can contains lemonade, shown shaded on the diagram in which XY is diameter. What fraction of the volume of the can is filled with lemonade:
just below a quarter; just over a quarter; exactly a quarter?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Image
Imagine the liquid became solidified.
Rotate the can until the face which contains XY is
horizontal.
Now take the horizontal cross-section half way up the can
which meets the plane face of the "liquid" in the line AB.
Observing that AB bisects the radius PQ, rotate the upper
section of the "liquid" about AB through 180 degrees. The new
position is shaded darkly and fits into the bottom right hand
quarter (with some space to spare).
The actual fraction can be shown to be approximately 0.21. The
answer is therefore less than a quarter.