Billy invites four friends over for a birthday party. His cake is made in the shape of a 10-inch by 10-inch square, one inch high, with frosting on the top and sides. How can the cake be cut with just five slices of the knife so that each of the five persons get a piece with same amount of cake and the same amount of frosting?
Solve the following time-rate-distance problems using the IPO method. Ignore time zones.
Output : The placement of the five cuts
Input : A 10 by 10-inch square one inch high, with frosting on top and sides Cut into five slices with five cuts of the knife. Each piece has the same amount of cake and the same amount of frosting
Approach Simplify the two conditions:
Condition 1: Same amount of cake in each piece
This requires the volumes must be equal. The height of each piece is the same.
Thus the top areas must be equal.
Condition 2: Same amount of frosting on each piece
This requires the top and side areas to be equal. (This is enforced in condition 1 above.)
Thus, the portion of the total perimeter given to each piece must be the same.
Thus, each piece must have the same area and the same amount of the total perimeter
If the cake were circular, these conditions could be met by cutting from the center to the edge so each piece has the same angle for its wedge. Use trial and error to see if a similar approach works for a square cake?
Before cutting the cake, find the center of the top at the point where the two diagonals cross. Then mark the 40-inch perimeter into one-inch divisions, ten per side.
Mark off 40 small triangles that radiate from the center to the edge. Even though the shapes of the triangles are not all the same, the base and altitudes are. Each triangle will have a base of 1/40 of perimeter (1 inch) and an altitude half a side (5 inches).
Cut the cake in five slices, starting with half a diagonal and continuing on the side of every eighth small triangle.
Test to see if the conditions above have been met.
Each piece will have eight small triangles, each with one inch of perimeter. Thus, the amount of perimeter for each piece will be eight inches.
Each piece will have a top area of eight small triangles, each with a base of one inch and an altitude of 5inches. Thus, the area of each piece will be:
Thus, both condition1 and condition have been met.
Check to see if the volume and frosting area of each piece is 1/5 of total for cake.
Cake volume:
The total amount of cake is the same as its volume (v):
Frosting Area:
The total amount (a) of frosting is the same as its top and side areas:
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