Problem of the Week
Problem
B and Solution
Equal
Cake and Icing
Problem
Serenity is having \(16\) guests for
dinner. She baked a cake for dessert using a square cake pan with side
length \(36\text{ cm}\). The cake is
\(8\text{ cm}\) tall. The top face and
side faces of the cake are covered in icing.
She would like to slice the cake into \(16\) pieces. She calls a slicing a "fair
cake" if each piece has the same amount (volume) of cake and the same
amount (surface area) of icing.
To cut the cake into \(16\)
pieces, suppose she makes three equally-spaced vertical slices and three
equally-spaced horizontal slices through the top face of the cake. Is
this a fair cake?
To cut the cake into \(16\)
pieces, suppose she first divides each edge of the top face into four
equal lengths. She then makes a straight slice from each end of a
length, through the centre of the square, to an end of a length on the
opposite edge. Is this a fair cake? Show calculations to support your
answer.
Extension:
Only \(9\) guests want to eat
dessert. Serenity decides to cut the cake into \(9\) pieces by dividing the entire perimeter
of the cake into nine equal lengths, starting in the top-left corner and
moving clockwise. She then makes a slice from each end of a length to
the centre of the square. Is this a fair cake? Show calculations to
support your answer.
Solution
Since the side length of the square pan is \(36\text{ cm}\), each slice has a square top
face with side length \(36\div 4 = 9\text{
cm}\).
Since the height of the cake is \(8\text{
cm}\), the volume of each slice is \(9\times 9\times 8 = 648\text{ cm}^3\).
Thus, each slice has the same volume of cake.
The top face of each slice has \(9\times 9
= 81\text{ cm}^2\) of icing. Each side face with icing will have
\(9\times 8 = 72\text{ cm}^2\) of
icing. Thus, the corner pieces will have \(81
+ 72 + 72 = 225\text{ cm}^2\) of icing, the edge pieces that are
not corner pieces will have \(81 + 72 =
153\text{ cm}^2\) of icing, and the middle pieces will have only
\(81\text{ cm}^2\) of icing.
Since each slice does not have the same amount of icing, this is not
a fair cake.
The top face of each slice is in the shape of a triangle. Since
the side length of the square pan is \(36\text{ cm}\), the base of each triangle
is \(36\div 4 = 9\text{ cm}\). The
height of each triangle is half of the side length of the square pan, or
\(36\div 2 = 18\text{ cm}\). Using the
formula for area of a triangle, we have that the area of the top face of
each slice is \(\text{base} \times
\text{height} \div 2 = 9 \times 18 \div 2 = 81\text{ cm}^2\).
Since each slice has the same top face area of \(81\text{ cm}^2\) and same height of \(8\text{ cm}\), each slice has the same
volume of \(81 \times 8 = 648 \text{
cm}^3\).
The top face of each slice has \(81\text{
cm}^2\) of icing. Since the base of each triangular top face is
\(9\text{ cm}\) and the height of each
slice is \(8\text{ cm}\), each slice
has a side face with \(9\times 8 = 72\text{
cm}^2\) of icing. Thus, each slice has the same amount of icing,
\(81 + 72 = 153 \text{ cm}^2\).
Since each slice has the same volume and the same area of icing, this
is a fair cake.
Solution to
extension:
Since the perimeter of the cake is \(36
\times 4 = 144\text{ cm}\) and \(9\) slices are made, then each piece will
have a total edge length of \(\frac{144}{9} =
16\text{ cm}\) with icing. Thus, since the height of the cake is
\(8\text{ cm}\), the amount of icing on
the side of each slice is \(16\times 8 =
128\text{ cm}^2\).
For six of the slices, the top face of the slice is a triangle. For
the remaining three slices the top face is a quadrilateral. These slices
are marked \(A\), \(B\), and \(C\).
First we look at the triangular slices. The base of each triangle is
\(16\text{ cm}\) and the height is half
the side length of the square pan, or \(36\div
2 = 18 \text{ cm}\). Using the formula for area of a triangle, we
have that the area of the top face of each triangular slice of cake is
\(\text{base} \times \text{height} \div 2 = 16
\times 18 \div 2 = 144\text{ cm}^2\).
Next we look at the quadrilaterals. Each quadrilateral consists of
two triangles, each with height \(18\text{
cm}\).
The top face of slice \(A\) has one
triangle with base length \(36 - 16 - 16 =
4\text{ cm}\). Thus, the other triangle has base length equal to
\(16-4 = 12\text{ cm}\). Therefore,
using the formula for the area of a triangle, we can determine that the
area of the top face of slice \(A\) is
\(4\times 18 \div 2 + 12\times 18 \div 2 = 36
+ 108 = 144\text{ cm}^2\).
The top face of slice \(B\) has one
triangle with base length \(36 - 12 - 16 =
8\text{ cm}\). Thus, the other triangle has base length equal to
\(16-8 = 8\text{ cm}\). Therefore,
using the formula for the area of a triangle, we can determine that the
area of the top face of slice \(B\) is
\(8\times 18 \div 2 + 8\times 18 \div 2 = 72 +
72 = 144\text{ cm}^2\).
The top face of slice \(C\) has one
triangle with base length \(36 - 8 - 16 =
12\text{ cm}\). Thus, the other triangle has base length equal to
\(16-12 = 4\text{ cm}\). Since these
are equal to the base lengths of the triangles in the top face of slice
\(A\), it follows that the area of the
top face of slice \(C\) is also \(144\text{ cm}^2\).
Therefore, since each slice has the same top face area of \(144\text{ cm}^2\) and the same height of
\(8\text{ cm}\), each slice has volume
equal to \(144 \times 8 = 1152\text{
cm}^2\).
Also, each slice has \(144\text{
cm}^2\) of icing on top and \(128\text{
cm}^2\) of icing on the side, for a total of \(144 + 128 = 272\text{ cm}^2\) of icing.
Since each slice has the same volume and the same area of icing, this
is a fair cake.
