A+ The exploration has a title

The optimization of the mixing time for Achieving the

Desired Chocolate Cupcake Dome

May 2025 Session

 A+ well explained introduction

Introduction:

Making chocolate cupcakes is a task that calls for accuracy and focus because several variables affect
the final product. The ratio of wet to dry components, which affects the cupcake's texture and
structure, is one important factor I observed throughout baking. The cupcakes' flavour and moistness
can be greatly impacted by additional variables including the type of cocoa powder used, oven
temperature, and mixing time.

Growing up, I watched baking videos and recall bakers claiming that the perfect chocolate cupcake
should be moist, have a smooth consistency, and taste strongly of chocolate. The exact techniques and
ingredient ratios needed to produce such outcomes are still unidentified despite this widespread
agreement. Among the specific questions my research aims to answer are: What effects does mixing
time have on the structural integrity and rise of cupcakes? What changes may be made to maximise
the texture and curvature of the dome? The goal of this study is to advance a more accurate
understanding of baking methods by methodically examining these variables. C+ Personal engagement

For me, baking has always involved more than just technical skills. I'm extremely proud of my hobby
as it allows me to connect with my heritage and express myself in a meaningful way. Growing up in
Oman, my family would regularly get together to talk about traditional foods. Cooking and baking
still hold a vital position in our everyday lives, even though cupcake creation is not a traditional
hobby. I watched for hours as my grandmother's hands worked organically, kneading fresh dough in a
slow, rhythmic way. She would tell me that this was more than just a way to prepare food; it was how
she found calm in the midst of the turmoil of the day.

I would like to explore these factors to improve my cupcake recipe because I love baking and am
always trying to get better. I want to improve my future baking attempts by better understanding the
elements that go into making a perfectly baked chocolate cupcake through experimenting.

Finding the exact ingredient ratios and baking methods required to make the ideal chocolate cupcake
is the aim of this study. This will be accomplished by investigating the effects of factors like the ratio
of wet to dry ingredients and the duration of mixing on the structure, texture, and flavour of the
cupcake. I aim to identify the changes needed to get the best possible outcome by contrasting my
cupcakes with a professionally acclaimed "perfect" chocolate cupcake.

Three main investigations serve as the framework for this investigation. Investigation I varied the
mixing periods and measured the height and curvature of the resulting domes to investigate the
structural characteristics of cupcakes. The rounded, curved top of a cupcake that develops during
baking is called a cupcake dome. It happens when steam and air expand in the oven, causing the batter
to rise. These measurements were used to establish mathematical relationships between mixing time
 A+
and cupcake morphology. In Investigation II, the focus shifted to modelling the characteristics of a
professionally recognized "perfect cupcake," utilizing vertex form equations and differentiation to
define the optimal dome parameters. Finally, Investigation III involved baking a new batch of
cupcakes based on the estimated optimal mixing time identified in the earlier investigations. The
results were then evaluated against the "perfect cupcake" model to determine the accuracy of the
prediction and the applicability of the findings.

Methodology

Three batches of cupcakes will be prepared using a consistent base recipe, with the only variable
being the mixing time. This variable is critical, as it directly affects the aeration of the batter and,
consequently, the rise and texture of the cupcakes. The mixing times I chose for this study are 2
minutes, 10 minutes, and 30 minutes. Each of the batches will undergo analysis where I will assess
dome height, curvature, and crumb structure.

To obtain accurate measurements, cross-sectional imaging will be utilized to capture the domes’
profiles. These measurements will be analyzed mathematically, with the curvature of each dome
modelled using quadratic equations. GeoGebra www.geogebra.org will be used for these calculations,
to enable precise representation of the domes' structure. I will then use Differentiation to identify
critical points, such as the vertex. The vertex of a curve is the point where it reaches its maximum or
minimum value, depending on the curve's shape. It represents the point where the slope of the tangent
is zero, meaning that the curve changes direction to determine the maximum height of the domes. This
method aligns with concepts from the IB Mathematics Applications and Interpretation syllabus,
specifically within the Functions and Calculus units, which include quadratic functions,
differentiation, and their practical applications.

To provide a clear benchmark for comparison, the professionally recognized standards outlined in
Professional Baking by Wayne Gisslen (Gisslen, 2021) will be used as a reference. This resource
offers detailed insights into the characteristics of high-quality baked goods, including cupcakes, and
will guide the evaluation of how closely the cupcakes align with the "perfect cupcake" standard

Finally, the results from the three mixing times will be analyzed to identify patterns and trends. The
study will forecast the mixing period that best strikes a balance between aeration and structural
integrity by calculating the curvature of the domes and analysing their physical characteristics. In
addition to improving baking process accuracy, this study attempts to measure the correlation between
cupcake shape and mixing time.

A+
The exploration has an elaborate introduction and aclear description of the aim and methodology
B+ use of meaningful mathematics language
The following section, Investigation I, will focus on examining cupcake domes influenced by different
mixing times. It outlines the methods used to measure and compare structural variations, connecting
these findings to the goal of determining the ideal mixing time for achieving the "perfect cupcake."
Through detailed modelling and comparison, this stage builds a clear understanding of how mixing
impacts the shape and height of the domes, serving as a basis for further analysis.

Investigation I: Analyzing the Structure of Chocolate Cupcakes with

Varying Ingredient Ratios

I will be analyzing and modelling the structure of 3 chocolate cupcakes for 3 different mixing times. I
made all the cakes using the same recipe, keeping the only independent variable as ‘mixing time’. The
dome will only be modelled that has risen beyond the initial pouring level of the batter. This level can
be seen clearly on the cupcake as it is the point at which the doming usually begins and the diagonal
line on the side of the cupcake ends.

The analysis will focus on the domed section of the cupcakes, which is the part that rises beyond the
initial level of the batter. This level is visible on each cupcake at the point where the batter initially
settled in the liner before baking. For this investigation, this level will be referred to as the "Batter
Line." The Batter Line will serve as the “x” axis of the model, while the “y” axis will indicate the rise
of the cupcake.

The "x" axis extends to 5 cm, which was the consistent diameter of the cupcakes after baking. Each
increment on the graph represents 1 cm to scale. All of the cupcakes were measured after baking to
accurately model their structure to scale on GeoGebra. This modelling will allow for a detailed
comparison of the doming patterns across the three mixing times.

The following images display the cross-sections of three cupcakes prepared with mixing times of (left
to right) 2 minutes, 10 minutes, and 30 minutes, and serve as a visual representation of the structural
differences that mixing time creates. It illustrates the physical outcomes of varying mixing times,
which are further quantified and analyzed mathematically.

Figure 1: Cross-section of all 3 cupcakes with different mixing times

B+ use of clear mathematical language
Modeling the Cupcake Domes Using Vertex Form

In this investigation, the doming of each cupcake will be plotted on GeoGebra using Vertex Form. The
Vertex Form equation has three distinct coefficients: a, h, and k. These coefficients influence aspects
of the parabola, including its shape, position, and its orientation. This form is beneficial for my
analysis as it provides direct information about the vertex of the function, which represents the highest
point of the cupcake dome.

I utilized GeoGebra to calculate the coefficients 𝑎, ℎ, 𝑘 for the domes of all three cupcakes, allowing
for accurate vertex-form modelling. To do this, the program was used to plot the curvature of the
measured data points for each cupcake's dome, which was taken from cross-sectional photographs.
GeoGebra’s regression tool was then used to fit a parabola to the data points, generating the vertex
2
form equation 𝑦 = 𝑎(𝑥 − ℎ) + 𝑘. The coefficient ℎ represents the horizontal position of the vertex,
𝑘 corresponds to the maximum height of the dome, and aaa determines the steepness and orientation
of the curve. Each step of the process, from plotting points to obtaining the equation, was
systematically carried out to ensure accuracy in this investigation.

GeoGebra can be accessible at www.geogebra.org for reproducibility and accessibility. To

demonstrate how the coefficients were identified and employed in the study, subsequent sections
provide a detailed explanation of the procedure used to model these equations. By using this
technique, I made sure that the domes were accurately represented numerically, enabling an extensive
analysis of cupcake structures over the course of the three mixing times.

Calculation for Cupcake Domes

2
To model the cupcake domes using the Vertex Form equation 𝑓(𝑥) = 𝑎(𝑥 − ℎ) + 𝑘, ​ the
coefficients 𝑎, ℎ, and 𝑘 were determined using the following process:

1. Collecting Data Points

-​ Using GeoGebra, cross-sectional images of the cupcakes were overlaid with a grid
-​ Multiple points along the dome’s curvature were identified and their coordinates (𝑥, 𝑦) were
recorded. These points represented the height of the dome at various horizontal positions.

2. Determining the Vertex (ℎ, 𝑘)

-​ The vertex (ℎ, 𝑘) of each parabola corresponds to the highest point of the dome. I identified
this point as the peak of the dome in the grid overlay:
-​ For the 10-minute mixing time: (3. 07, 2. 00)
B+ defined terminologies and correct notations
 -​ For the 2-minute mixing time: (3. 28, 0. 75)
-​ For the 30-minute mixing time: (3. 21, − 0. 90)

3. Calculating the Coefficient 𝑎

2
-​ Using the vertex form equation 𝑓(𝑥) = 𝑎(𝑥 − ℎ) + 𝑘, another point from the dome’s
curvature was substituted into the equation to solve for 𝑎

Example calculation for 10 minutes mixing time:

-​ Vertex: (3. 07, 2. 00)

-​ Additional point: (4. 07, 1. 72)
2
-​ Substituting: 1. 72 = 𝑎(4. 07 − 3. 07) + 2. 00
2
-​ Simplify: 1. 72 = 𝑎(1) + 2. 00
-​ Solving for a: 𝑎 =− 0. 28

The same process was repeated for the 2-minute and 30-minute mixing times:

-​ 2-Minute Mixing Time: 𝑎 =− 0. 10

-​ 30-Minute Mixing Time: 𝑎 = 0. 22

4. Verifying the Model E+ mathematic used support development of the exploration

-​ The full vertex form equations were constructed for each cupcake:
2
-​ 2 minutes: 𝑓(𝑥) =− 0. 10(𝑥 − 3. 28) + 0. 75
2
-​ 10 minutes: 𝑓(𝑥) =− 0. 28(𝑥 − 3. 07) + 2. 00
2
-​ 30 minutes: 𝑓(𝑥) = 0. 22(𝑥 − 3. 21) − 0. 90

I plotted these equations using GeoGebra to confirm they accurately represented the domes’
curvatures

A+ well explained sub titles

Cake Mixing time Graph Vertex Form: 𝑓(𝑥)= 𝑎 (𝑥 − ℎ) + 𝑘
2

2 Minutes 2
𝑓(𝑥) =− 0. 10(𝑥 − 3. 28) + 0. 75

𝑎 =− 0. 10
ℎ = 3. 28
𝑘 = 0. 75

Vertex: (3. 28, 0. 75)

10 Minutes 2
𝑓(𝑥) =− 0. 28(𝑥 − 3. 07) + 2. 00

𝑎 =− 0. 28
ℎ = 3. 07
𝑘 = 2. 00

Vertex: (3. 07, 2. 00)

30 Minutes 2
𝑓(𝑥) = 0. 22(𝑥 − 3. 21) − 0. 90

𝑎 = 0. 22
ℎ = 3. 21
𝑘 =− 0. 90

Vertex:(3. 21, − 0. 90)

B+ &E+ use of graphs and relevant mathematics

Analysis: Accuracy of the Calculations and Models

2-Minute Mixing Time:

2
-​ The model𝑓(𝑥) =− 0. 10(𝑥 − 3. 28) + 0. 75 represents the general shape of the cupcake
dome, but it lacks precision in capturing its curvature. While the vertex (3. 28, 0. 75) reflects
the peak height accurately, the sides of the dome deviate from the ideal parabolic shape. This
discrepancy suggests that insufficient mixing resulted in uneven aeration, leading to a less
defined dome structure. The model is less effective in capturing the subtleties of the
under-mixed batter’s irregular curvature.

10-Minute Mixing Time:

2
-​ The model𝑓(𝑥) =− 0. 28(𝑥 − 3. 07) + 2. 00 closely aligns with the observed dome, with
the vertex (3. 07, 2. 00) matching the peak of the cupcake. This trial shows the most precise
representation, as the dome’s structure aligns to a parabolic shape. The stable rise of the batter
indicates that the 10-minute mixing time achieved an optimal balance between aeration and
structural integrity, making this model highly reliable.

30-Minute Mixing Time:

2
-​ The model 𝑓(𝑥) = 0. 22(𝑥 − 3. 21) − 0. 90 effectively represents the collapsed structure
of the dome, with the vertex (3. 21, − 0. 90) accurately capturing the lowest point. However,
the edges of the dome deviate significantly from the model, reflecting the irregularities that
were caused by overmixing. Excessive mixing might have destabilized the batter, weakening
the dome's ability to maintain its form. While the model provides a general representation, it
is less accurate in capturing the full extent of the structural inconsistencies.

The most accurate model was produced by mixing for 10 minutes since the dome closely resembles a
parabolic curve. Under-mixing and over-mixing, respectively, resulted in abnormalities that generated
larger deviations in the 2-minute and 30-minute trials. This demonstrates how difficult it is to use a
single quadratic model to represent non-ideal dome shapes. The distortions seen in the under-mixed
and over-mixed experiments are not fully taken into account by the vertex form equation, although it
depicts symmetrical and well-formed domes quite well. Notwithstanding these limitations, the models
show how mixing duration and dome shape relate to one another, offering insight into how mixing
affects structural integrity. D+ reflection
Verification of Vertex Form Using Standard Form

The process of expanding the vertex form into standard form and verifying it using differential
calculus has a critical role in ensuring the accuracy and validity of the mathematical model used to
represent the cupcake dome. By converting the equation from one form to another, it acts as a
verification mechanism, providing confidence that the mathematical representation aligns with the
observed data. I can check my answer by expanding the function into standard form and then use
differential calculus to check my vertex. Additionally, for irregular shapes, such as the dome from the
30-minute mixing time, this step highlights the model's limitations and the potential need for more
complex equations.

This is what the standard form looks like:

2
𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐

The first step is to expand everything within the bracket. After this, the values outside the brackets are
distributed and all like terms are added and organized in order of degree.

To expand, I used Standard Identity II, which states:

2 2 2
(𝑎 − 𝑏) = 𝑎 − 2𝑎𝑏 + 𝑏

To verify the vertex form function for the 30-minute mixing time, I expanded it into standard form
and confirmed its accuracy. The vertex form equation is:

2
𝑓(𝑥) = 0. 22(𝑥 − 3. 21) − 0. 90

Expanding the equation:

2 2 2
Using the identity: (𝑎 − 𝑏) = 𝑎 − 2𝑎𝑏 + 𝑏

2 2
𝑓(𝑥) = 0. 22(𝑥 − 2(3. 21)𝑥 + (3. 21) ) − 0. 90

2
𝑓(𝑥) = 0. 22(𝑥 − 6. 42𝑥 + 10. 3041) − 0. 90

Distributing 0.22 across the terms:

2
𝑓(𝑥) = 0. 22𝑥 − 1. 4124𝑥 + 2. 266902 − 0. 90

Combining the constants:

2
𝑓(𝑥) = 0. 22𝑥 − 1. 4124𝑥 + 1. 366902
This matches the quadratic standard form:

𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

B+ Key terms and Variables

Where: defined and use of correct
notations
-​ 𝑎 = 0. 22
-​ 𝑏 =− 1. 4124
-​ 𝑐 = 1. 366902

𝑑𝑦
To find the vertex, I can differentiate the standard form equation and then set 𝑑𝑥
= 0 to find the

value of 𝑥

After finding 𝑥, I need to substitute it back into the standard form to find 𝑦

Differentiating:

𝑑𝑦
𝑑𝑥
= − 0. 44𝑥 − 1. 4124

𝑑𝑦
Set 𝑑𝑥
= 0:

0 =− 0. 44𝑥 − 1. 4124

1.4124
𝑥= 0.44

𝑥 = 3. 21 (2 d.p.)

Substitute 𝑥 = 3. 21 into 𝑓(𝑥)

2
​ ​ 𝑓(𝑥) = 0. 22(3. 21) − 1. 4124(3. 21) + 1. 366902

​ ​ 𝑓(𝑥) = 0. 22(10. 3041) − 1. 4124(3. 21) + 1. 366902

𝑓(𝑥) = 2. 266902 − 4. 534804 + 1. 366902

𝑓(𝑥) =− 0. 90 (2 d.p.)

Using the GDC I have found the 𝑥 value of the vertex, and I see that it matches.

Substituting the derived x-value back into the standard form equation verified the corresponding
y-value, to ensure that the calculations are aligned with the vertex form equation. This process
validates the accuracy of the model and demonstrates consistency between the two forms of the
quadratic function:
 2
𝑓(𝑥) = 0. 22𝑥 − 1. 4124𝑥 + 1. 3669023

The vertex was confirmed at (3.21,−0.90), which matches the peak point in the vertex form equation,
further strengthening its reliability for analyzing dome structure. By comparing the domes across
different mixing times, the vertex becomes a key parameter for quantitative analysis of the rise and
structural symmetry, directly contributing to the exploration's goal of understanding how mixing time
affects cupcake morphology.
E+ Use of mathematics that is relvenat and demostrates reasoning

Reflection I: Limitations and Observations

The lack of a completely domed quadratic curve in any of the cupcakes highlights a limitation in this
analysis technique as it fails to adequately account for the range of non-quadratic geometries. This
suggests that variables such as ingredient proportions, oven temperature, and baking time may play
crucial parts in determining the shape and texture of the dome, which requires more research for better
clarity. The applied equations have been designed to be as accurate as possible. For assessing the
consistency and rise of the cupcake's structure, they offer important information about critical
elements such as peak height and the dome's shape balance.

Only one variable was changed during the baking and mixing processes to ensure consistency.
However, the findings imply that alternative modelling methods, such as more specialised methods for
different textures or higher-degree polynomial curves, may more accurately capture these changes.
Furthermore, areas that are flatter might fit linear models better.

To build upon this analysis, comparing the cupcake curves with their corresponding equations is
essential. Such comparisons could benefit from including overlooked factors, such as ingredient
textures and the environmental conditions during baking, to deepen the understanding of these
outcomes. These comparisons are detailed below, tying into the framework of insights developed
earlier in this analysis.
D+ Critical reflection
evident and C+ indipendent
Function Data Analysis for Mixing Times thinking

The following quadratic equations represent the cupcake domes for the three different mixing times:

2
2 minutes: 𝑓(𝑥) =− 0. 10(𝑥 − 3. 28) + 0. 75

2
10 minutes: 𝑓(𝑥) =− 0. 28(𝑥 − 3. 07) + 2. 00

2
30 minutes: 𝑓(𝑥) = 0. 22(𝑥 − 3. 21) − 0. 90

As observed from the function data, the trial mixed for approximately 2 minutes displayed the least
positive rise, indicating minimal air incorporation within the batter. This is reflected in the low 𝑘
-value of 0.75, which corresponds to the maximum point of the graph. The cake barely rose, achieving
a height of just 0.75 cm.

In contrast, the 10-minute trial achieved a significant rise, with a 𝑘-value of 2.00, indicating a height
of 2.00 cm. This substantial increase demonstrates the importance of adequate mixing in incorporating
air into the batter.

However, the 30-minute trial demonstrates a significant drop, with a k-value of -0.90, signifying the
cake's fall to a height of -0.90 cm. This implies that too much air was added during mixing,
weakening the structure and resulting in its collapse after baking.

The cakes consistently peaked at the centre, as indicated by the fact that every trial had the same
h-value, which represents the vertex's x-coordinate. The function's width, or a-value, changed from
trial to trial. The larger, symmetric rise and fall patterns of the 10- and 30-minute trials resulted in
equal widths, whereas the 2-minute trial's shorter breadth was caused by its minimum rise.

Since there is no sufficient curvature to support modelling with a quadratic function, the 2-minute
trial's small rise raises the possibility that a linear equation could more accurately represent the
information. This finding emphasises the drawbacks of using quadratic models in minimal-rise
situations and the significance of choosing the right mathematical models for various physical results.

Investigation II: Modelling the ‘Perfect Cupcake’

The definition of the "perfect cupcake" was established by referencing professional baking standards.
These guidelines highlight three essential qualities of the perfect cupcake: consistent crumb
distribution, symmetrical dome curvature, and structural stability. A professionally recognised
"perfect cupcake," with its dome shape and crumb texture, is shown in Figure 2. The basis for
comparing the outcomes of my cupcake experiments is this visual reference. Its presence emphasises
the foundation for figuring out how quantitative modelling and qualitative evaluations of cupcake
morphology agree. This lays out the justification for further computations and comparison analyses.
 Figure 2: The ‘perfect cupcake’

Using the same methodology as the earlier analysis, I looked at the features of my chocolate cupcake.
To construct mathematical parameters for the "perfect cupcake," I employed quadratic modelling
approaches, which were akin to those utilised in the earlier study. Vertex form equations were used to
analyse the curvature of the dome, allowing key parameters like the vertex (h, k) to be calculated.
While parameter "a" measures the curvature's symmetry and steepness, these values indicate the
dome's maximum height and horizontal placement, respectively. The modelled "perfect cupcake" and
the experimental cupcakes can be directly compared thanks to this research, which also highlights any
variations brought on by different mixing times.

The parabolic model applied to the "perfect cupcake" dome using GeoGebra software is shown in
Figure 3. This representation captures the structural curvature of the cupcake and highlights the apex
of the dome, which corresponds to the vertex of the quadratic function. By analyzing this graph, key
factors such as the maximum height and symmetry of the dome are determined, providing a basis for
further comparisons with cupcakes subjected to varied mixing times.

B+ well labelled and appropriate graphs

​ Figure 3: ‘Perfect cupcake’ modelled with a parabola using GeoGebra

After modelling my graph, I have found the function that describes it, shown below:

2
​ ​ ​ ​ 𝑓(𝑥) =− 0. 226𝑥 + 1. 47𝑥 − 1

To get this function in vertex form, the following steps were carried out:

The general standard form of a quadratic function is:

2
𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐

Convert the Standard Form to Vertex Form:

The vertex form of a quadratic equation is:

2
𝑦 = 𝑎(𝑥 − ℎ) + 𝑘

where (ℎ, 𝑘) is the vertex of the parabola

2
1. Factor out the coefficient of 𝑥 from the first two terms:

2 1.47 E+ use of mathematics

𝑦 =− 0. 226(𝑥 − 0.226
𝑥) − 1 that demostrates
reasoning with evidence

Simplify:

2
𝑦 =− 0. 226(𝑥 − 6. 50𝑥) − 1

2. Complete the square inside the parentheses: Take half the coefficient of 𝑥 = (− 6. 50), square
it, and add and subtract it inside the parentheses:

2 −6.50 2 −6.50 2
𝑦 =− 0. 226(𝑥 − 6. 50𝑥 + ( 2
​) − ( 2
​) ) − 1

2
𝑦 =− 0. 226(𝑥 − 6. 50𝑥 + 10. 5625 − 10. 5625) − 1

3. Group the perfect square trinomial:

2
𝑦 =− 0. 226((𝑥 − 3. 25) − 10. 5625) − 1

4. Distribute the − 0. 226 value:

2
𝑦 =− 0. 226(𝑥 − 3. 25) + (− 0. 226)(− 10. 5625) − 1

2
𝑦 =− 0. 226(𝑥 − 3. 25) + 2. 39 − 1
5. Simplify to get vertex form:

2
​ 𝑦 =− 0. 226(𝑥 − 3. 25) + 1. 39

The vertex indicates the highest point of the modelled cupcake dome. Here, the vertex (3. 25, 1. 39)
suggests that the dome reaches a maximum height of 1.39 cm at 𝑥 = 3. 25, which is the centre of the
cupcake's width.

Now that I have found the standard form, I must differentiate and equate it to 0 to find the value 𝑥 to
identify the vertex of the parabola, which corresponds to the highest point of the cupcake dome.
Differentiation provides the slope of the function at any given point, and a slope of zero indicates a
horizontal tangent, marking the peak of the dome. This method makes it possible to identify the
vertex's x-coordinate, which denotes the location where the dome reaches its highest point. I can
accurately examine the cupcake dome's geometric characteristics by computing this value, and I can
then utilise the results for additional comparisons and assessments throughout the investigation:

𝑑𝑦
​ ​ ​ ​ 𝑑𝑥
= − 0. 452𝑥 + 1. 47

𝑑𝑦
Set 𝑑𝑥
= 0: ​

​ ​ ​ ​ 0 =− 0. 452𝑥 + 1. 47
E+ use of mathematics that demostrates reasoning
with evidence
1.47
𝑥= 0.452
≈ 3. 25 (2 d.p)

Now I’m going to substitute 𝑥 = 3. 25 back into the standard form to find 𝑦:

2
𝑓(3. 25) =− 0. 226(3. 25) + 1. 47(3. 25) − 1

𝑓(3. 25) ≈ 1. 39 (2 d.p)

The value 𝑦 also matches, confirming that the vertex found is accurate. Calculating the vertex
establishes the maximum height and its position, providing a key reference for comparing dome
structures across mixing times. This calculation supports the evaluation of mixing duration's impact
on cupcake morphology and informs the identification of the optimal conditions for achieving the
desired dome shape.

Reflection II: Comparison to Original Cupcakes

Considering that I have now determined the equations for all my cupcakes, I can attempt to estimate
the ideal mixing time required to produce a "perfect" cupcake. If I had access to more data points, I
could create a more detailed equation to model the relationship between mixing time and the
cupcake's rise. However, with only three data points, I must rely on an educated estimate.

The "perfect cupcake" had a vertex of (3. 25, 1. 39), with the 𝑥-coordinate closely matching that of
my chocolate cupcake but differing slightly in the 𝑦-value. This suggests that mixing time primarily
influences the height of the dome (maximum point).

The closest comparison to a "perfect cupcake" is my 10-minute chocolate cupcake, which had a vertex
of (3. 07, 2. 00). The 𝑦-coordinate aligns precisely with the expected height of the "perfect cupcake."
Based on this analysis, I estimate that the optimal mixing time for achieving the "perfect cupcake"
would remain close to 10 minutes, confirming the symmetry and rise as ideal.
D+
Reflections
Investigation III: I baked another cupcake, this time with my estimated
mixing time gathered from my previous data. This was the result:

In this phase of my investigation, I aimed to validate the estimated optimal mixing time derived from
the analysis of the "perfect cupcake." Using the calculated mixing time, I baked another cupcake to
determine whether the theoretical findings could reliably replicate the desired dome structure and
texture. The resulting cupcake, shown in Figure 4, was analyzed and modelled to compare its
characteristics against those of the "perfect cupcake."

​ ​

Figure 4: Recreated cupcake with estimated mixing time​

This figure serves as a practical evaluation of the model's precision. By examining the recreated
cupcake both visually and mathematically, I assessed how effectively the estimated mixing time
replicated the dome's curvature and height. This process confirmed whether the insights gained from
earlier analyses could be reliably applied and reproduced in practice. Additionally, aligning the axes
and values during modelling ensured consistency, enabling a precise comparison between the
recreated cupcake and the reference "perfect cupcake," thereby strengthening the credibility of my
approach and results.
I modelled this cupcake with GeoGebra so I could compare my results to a "perfect cupcake." While
modelling, I followed the same steps as before to align my axis and values.​ ​ ​
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
​ ​ ​ ​ ​ ​ ​ ​

Figure 5: Modelling recreated cupcake

After modelling the cupcake, I found the function that describes it, shown below:
2
𝑓(𝑥) =− 0. 111𝑥 + 1. 114𝑥 − 1. 473

The equation is in standard form, and I converted it into vertex form to identify the vertex. Using the
𝑏
formula ℎ =− 2𝑎
​:
1.114
​ ​ ​ ​ ​ ℎ =− 2(−0.111)​

ℎ = 5. 02 (2 d.p)

Substituting ℎ = 5. 02 back into the standard form equation using the GDC to find 𝑘:
2
𝑘 =− 0. 111(5. 02) + 1. 114(5. 02) − 1. 473
B+ well labelled graph
E+ evidence of reasoning , knowledge and use fof
𝑘 ≈ 2. 33 (2 d.p)
technology

Thus, the vertex form of the equation is:

2
𝑓(𝑥) = − 0. 111(𝑥 − 5. 02) + 2. 33

The vertex of this function is approximately (5. 02, 2. 33).

To confirm the vertex, I calculated the derivative of the standard form and equated it to zero:

𝑑𝑦
​ ​ ​ ​ 𝑑𝑥
= − 0. 222𝑥 + 1. 114

𝑑𝑦
Setting 𝑑𝑥
= 0: ​
 0 =− 0. 222𝑥 + 1. 114

1.114
𝑥= 0.222
≈ 5. 02 (2 d.p)

Now substitute 𝑥 = 5. 02 back into the standard form to find 𝑦:

​
2
𝑓(5. 02) =− 0. 111(5. 02) + 1. 114(5. 02) − 1. 473

𝑓(5. 02)≈2. 33 (2 d.p)

The vertex (5. 02, 2. 33) that I found by using the GDC confirms the accuracy of the modelled
function. This calculation verifies the vertex of the cupcake’s dome, identifying the highest point on
the modelled curve. By finding the derivative of the standard form equation and setting it equal to
zero, the x-coordinate of the vertex is calculated. This method ensures that the derived vertex form is
accurate and aligns with the expected results. Confirming this value is essential for validating the
mixing time and its role in achieving the optimal dome shape, ensuring the conclusions drawn from
the investigation are consistent and applicable.

Reflection III: Evaluating success:

The comparison of the y-coordinates highlights how different mixing times affected the dome height
of the "perfect cupcake," which had a y-coordinate of 1.39. A 2-minute mixing time resulted in a
y-coordinate of 0.75, leading to a flatter dome due to insufficient air being incorporated into the batter.
With 10 minutes of mixing, the y-coordinate increased to 2.00, producing a taller dome, indicating
better air incorporation. However, 30 minutes of mixing caused the y-coordinate to drop to -0.90,
reflecting a collapsed dome likely caused by over-mixing, which weakened the batter's structure. This
comparison illustrates how mixing time directly impacts dome height and shows the importance of
precise timing to replicate the "perfect cupcake."

To evaluate how close my improved cupcake was to the "perfect cupcake," I calculated the percentage
error for all mixing times based on the 𝑦-value to check the change in rise of cupcakes.
D+ use of reflection through every step
The formula for percentage error is the following:

Where:​​
ε = 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑒𝑟𝑟𝑜𝑟

𝑉𝐴​ = 𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

𝑉𝐸 ​ = 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒

The brackets, indicating what we call “absolute value” are used to ensure that the percentage error
reflects only the size of the difference between the actual and expected values, without regard to
whether the actual value is higher or lower.

1.​ For 2 minutes

0.79−1.39
ε = || 1.39 || x 100%

−0.64
ε = || 1.39
|x 100%
|

ε = 46.04% (2 d.p)

2.​ For 10 minutes

2.00−1.39
ε = || 1.39 || x 100%

0.61
ε = || 1.39 ||x 100%

ε = 43.88% (2 d.p)

3.​ For 30 minutes

−0.90−1.39
ε = || 1.39 || x 100%

−2.29​ |
ε = || 1.39 |
x 100%

ε = 164.75% (2 d.p)
E+ Mathematics used is within the syllabus and error free

The results of the percentage error calculations provided insights into the relationship between mixing
time and dome height. The 10-minute mixing time showed the smallest percentage error of 43. 88%,
indicating it came closest to matching the ideal 𝑦-value of 1. 39. This suggests that 10 minutes of
mixing allowed for sufficient air incorporation, leading to a dome height more aligned with
expectations.
 D+ meaningful reflection

The 2-minute mixing time resulted in a percentage error of 46. 04%, slightly higher than the
10-minute trial. This indicates that under-mixing limited air incorporation, resulting in a smaller dome
height that still deviated significantly from the target value.

The 30-minute mixing time showed the largest percentage error at 164. 75%, highlighting the greatest
deviation from the "perfect cupcake." The over-mixing might have destabilized the batter, introducing
too much air and causing the dome to collapse, as reflected in the negative y-value.

The 10-minute mixing time proved to be the most effective in approaching the desired dome height.
These results highlight the significance of using an exact mixing time to strike a balance that reduces
variations and preserves the cupcake's structural integrity.

Conclusion A+ Logically developed hence coherent from the title to conclusion

This investigation demonstrated how mixing time significantly affects cupcake rise and structural
properties. Using mathematical modelling, I found that a 10-minute mixing period produced results
that were most similar to the "perfect cupcake," as shown by the vertex values matching the desired
parameters. The procedure showed the accuracy of my selected techniques in addition to quantifying
the correlation between cupcake morphology and mixing duration. The robustness of the analytical
technique was highlighted by the verification I carried out using both software-based tools and manual
distinction, which validated the findings' validity and reliability. These findings highlight how crucial
regulated mixing times are to getting reliable and appealing baking results.

Strengths of investigation

-​ Quantitative Framework: Utilizing mathematical functions and calculus allowed for a

systematic examination of dome curvature, producing accurate and measurable results. This
approach allowed for clarity and depth when analyzing the effects of mixing times on cupcake
structures.
-​ Integration of Technology: The application of GeoGebra streamlined the modelling process,
making it efficient and visually informative. Verifying these results with manual calculations
using the GDC further reinforced the reliability of the findings, ensuring consistency across
different methods.
-​ Practical Relevance: This investigation effectively connected abstract mathematical ideas
covered in the curriculum with an actual situation. My work illustrated the usefulness of
mathematical methods in tackling real-world problems by using theoretical tools to examine
the cupcake domes.

D+ reflections are critical

Limitations and Improvements

-​ Restricted Sampling: A deeper understanding of the connection between mixing time and
dome formation might be possible if the range of mixing times were extended to include other
intervals, such as 5- and 15-minute mixing durations.
-​ Environmental Variables: Potential discrepancies, such as unequal oven temperatures or slight
differences in ingredient measurements, might have affected my results. Establishing greater
control over these factors could improve the investigative experiment's accuracy and reduce
variability.
-​ Model Representation: The small rise seen during the 2-minute experiment was difficult for
the parabolic model to adequately represent. Better representations of particular results may
be obtained by incorporating alternate techniques, such as higher-degree polynomial
equations for complicated patterns or linear functions for limited-rise scenarios.​

Suggestions for Further Study

To better understand their combined impacts on the structure and possibly the texture of the cupcakes,
I could expand this analysis in future research by adding more factors, including the ratio of wet to
dry ingredients or baking temperatures. Using three-dimensional modelling software to examine dome
symmetry could also yield more in-depth information about structural differences.
References:

Gisslen, W. (2021). Professional baking (8th ed.). Wiley. Retrieved from

https://www.wiley.com/en-us/Professional+Baking%2C+8th+Edition-p-00097007

Jack and Beyond. (2024, April 8). The perfect cupcake. Retrieved from
https://jackandbeyond.com/blogs/cake-resources/the-perfect-cupcake
A+ Reference visible