Stage 1 Mathematics Methods B
Assessment Type 2: Mathematical Investigation
Introduction to Differential Calculus – Cake Tin
Introduction
, A cake tin manufacturer will be making cake tins ranging in size from “tiny” to “gigantic”. Some tins will
be square based, others will be rectangular based. In all cases, the manufacturer wants to maximise the
volume of each cake tin. This investigation focuses on the size of a square cut into a piece of tinplate to
form an open top cake tin of maximum volume. The optimum size of these squares will be determined
using various mathematical concepts and calculation, mainly including the laws of calculus and the
quadratic formula. Conjectures between the dimensions of the squares and the overall pieces of
tinplate will also be found and proven.
Part A
An open top cake tin is to be made by cutting a square from each corner of a square piece of tinplate
with side lengths l cm. Once the cut is made the sides are folded to form an open top cake tin. Let x cm
be the side length of the square cuts to be made. These dimensions of the tinplate along with the folded
cake tin are presented in figures 1 and 1A below respectively.
Figure 1
Figure 1A
l cm
x cm
l−2 x cm
l−2 x cm
x cm
l cm
If l cm is substituted with a value, an equation can be formed which relates the volume of the cake tin
to the variable x cm. For instance, if l=5 cm ;
Volume of a cuboid ( V )=length (l ) × width ( w ) × height (h)
If the length and width of the square tinplate are both 5 cm, these dimensions of the folded cake tin are
5−2 x cm, as the 2 squares of x cm are cut from each of these sides of the tinplate. This also means
that the height of the cake tin would be x cm, according the side of the tinplate which is folded in. These
dimensions of the cake tin can be seen in figure 1A above.
Now for the cake tin, V ( x )=l× w × h
l=5−2 x cm
w=5−2 x cm
h=x cm
∴V ( x )=( 5−2 x ) ( 5−2 x ) ( x )
¿ x ( 25−20 x +4 x 2 )
2 3
¿ 25 x−20 x + 4 x
Assessment Type 2: Mathematical Investigation
Introduction to Differential Calculus – Cake Tin
Introduction
, A cake tin manufacturer will be making cake tins ranging in size from “tiny” to “gigantic”. Some tins will
be square based, others will be rectangular based. In all cases, the manufacturer wants to maximise the
volume of each cake tin. This investigation focuses on the size of a square cut into a piece of tinplate to
form an open top cake tin of maximum volume. The optimum size of these squares will be determined
using various mathematical concepts and calculation, mainly including the laws of calculus and the
quadratic formula. Conjectures between the dimensions of the squares and the overall pieces of
tinplate will also be found and proven.
Part A
An open top cake tin is to be made by cutting a square from each corner of a square piece of tinplate
with side lengths l cm. Once the cut is made the sides are folded to form an open top cake tin. Let x cm
be the side length of the square cuts to be made. These dimensions of the tinplate along with the folded
cake tin are presented in figures 1 and 1A below respectively.
Figure 1
Figure 1A
l cm
x cm
l−2 x cm
l−2 x cm
x cm
l cm
If l cm is substituted with a value, an equation can be formed which relates the volume of the cake tin
to the variable x cm. For instance, if l=5 cm ;
Volume of a cuboid ( V )=length (l ) × width ( w ) × height (h)
If the length and width of the square tinplate are both 5 cm, these dimensions of the folded cake tin are
5−2 x cm, as the 2 squares of x cm are cut from each of these sides of the tinplate. This also means
that the height of the cake tin would be x cm, according the side of the tinplate which is folded in. These
dimensions of the cake tin can be seen in figure 1A above.
Now for the cake tin, V ( x )=l× w × h
l=5−2 x cm
w=5−2 x cm
h=x cm
∴V ( x )=( 5−2 x ) ( 5−2 x ) ( x )
¿ x ( 25−20 x +4 x 2 )
2 3
¿ 25 x−20 x + 4 x
