Optimization of baking time in volume of

sponge cake

International Baccalaureate Mathematics Analysis and Approaches

Internal Assessment Project

May 6, 2025

Total Pages: Put the total number of pages here

Introduction ....................................................................................................................................1
Data Collection ...............................................................................................................................2
Data Analysis and Interpretation .................................................................................................3
Reflection ........................................................................................................................................4
Conclusion ......................................................................................................................................4
Bibliography ...................................................................................................................................4
Appendix .........................................................................................................................................4
Introduction

Exploring how seemingly obscure mathematics manifests itself in my daily life has always

fascinated me. As an aspiring chemist, I chose to pursue ‘chemistry in baking’ as the topic of my

Personal Project in Grade 10. In the process, I found that much of the chemistry involved to

produce the “optimal cake” was intertwined with advanced mathematics, which I didn’t possess

enough knowledge of at the time. Therefore, I figured that giving baking another go would be

fitting, but this time through a mathematical lens instead now that I have acquired the knowledge

& skills necessary to do so. For my Personal Project, I set the goal of creating recipes from

scratch purely through trial-and-error for various baked goods, one being a sponge cake. This

included determining how long the cake would need to bake for. However, I found that I was

wasting both ingredients and time by blindly baking cakes at various temperatures without any

valid calculations to justify the chosen “optimal” temperature. Therefore, I thought it would be

appropriate to continue this pursuit of the temperature (or, at least, a range) that would yield the

optimal sponge cake.

Therefore, the aim of this exploration is to determine the temperature range that produces a

maximum cake volume by modeling the height of the cake over time as it bakes in the oven. The

maximum cake volume is a logical indicator and operationalization of an optimal sponge cake as

it is quantifiable. Given the height of the cake, I would be able to calculate the volume. The

highly controlled nature of this exploration with specific conditions calls for primary data

collection (more on this later). I initially predicted for the height of the cake to increase linearly

as time spent baking in the oven increases, which means the volume would also increase
proportionally (calculated using the formula for the volume of a cylinder, 𝑉 = 𝜋𝑟 2 ℎ, given the

height). However, upon conducting secondary background research, I found that baking a sponge

cake can be broken down into 3 stages: expansion, setting and browning. Although the cake

height does increase somewhat linearly as time passes during the expansion stages (as per my

initial prediction), it reaches its maximum height during the setting stage as it rises into its

permanent shape and structure (Connelly). I now predict the optimal time range to be within the

setting stage, now that I know the cake height won’t keep increasing forever. This research was

confirmed by a preliminary trial that I conducted, where I baked a single cake to observe how it

would rise before conducting my full primary data collection. To my surprise, I found that the

cake didn’t start rising until 15 minutes into baking in the oven; when it did rise, it formed a

near-perfect dome/hemisphere shape (see Image 1), instead of rising evenly throughout as I

anticipated (see Image 2). I then decided calculating the cylindrical volume would no longer be

logical, as the cake did not retain its cylindrical shape as it rose. I instead will be using solids of

revolution to calculate the volume via integration as a dome shape can be produced when

revolving a function around the x-axis.

Image 1

Image 2
Data Collection

I then proceeded to gather raw data via experimentation as primary research. I conducted 6 trials,

meaning 6 sponge cakes (3 were baked at a time as I only had 3 identical cylindrical molds).

Although it would have been ideal to perform at least 10 trials, I was already using an excessive

amount of ingredients and didn’t want anymore to go to waste; therefore, 6 trials would be

sufficient to calculate an average height. I made sure to use the same amount of pre-made cake

mix for all trials for the batter to contain the same ingredient ratio across all trials. The same

cylindrical mold was used for all trials to ensure that the initial volume of all cakes would be the

same and only differ by the dome volume once it rises. The batter filled up approximately 3/4 of

the cylindrical molds and was placed into an oven preheated to 180°C. The cakes were baked for

65 minutes in the oven, in which I took pictures of the dome at intervals of 5 minutes, trying my

best to maintain a parallel camera angle to the side of the mold. I then uploaded each photo to the

software Logger Pro, where I performed photo analysis to determine the height of the dome

given the height of the cake mold (8 cm). This was done for each trial at each time interval,

producing Table 1 (raw data).

Table 1: Height of cake dome during increasing time intervals baking in the oven
Time baking in the oven (minutes)

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Trial
1 0 0 0 0.90 1.46 2.19 2.98 3.85 3.65 2.87 2.59 2.53 2.46 2.41

Heigh Trial
t of 2 0 0 0 0.95 1.54 2.11 2.69 3.38 3.46 2.68 2.45 2.22 2.20 2.21
cake
dome Trial
(cm) 3 0 0 0 0.63 1.38 1.76 2.57 3.30 3.12 2.39 2.33 2.21 2.18 2.17

Trial
0.72 1.55 2.14 2.81 3.79 3.62 2.84 2.52 2.38 2.30 2.26
4 0 0 0
 Trial
0.88 1.36 1.93 2.56 3.45 3.28 2.63 2.47 2.15 2.07 2.05
5 0 0 0

Trial
0.69 1.49 2.02 2.77 3.34 3.17 3.15 2.31 2.09 2.06 2.00
6 0 0 0

The average of all trials for each time interval was then calculated. From this, I can draw the
conclusion that the greatest average height was reached at minute 35.

Table 2: Average height of cake dome during increasing time intervals baking in the oven
Time baking in the oven (minutes)

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Average
Height of
0 0 0 0.9 1.46 2.19 2.98 3.85 3.65 2.87 2.59 2.53 2.46 2.41
cake dome
(cm)

Using the average heights recorded above, integration can be used to find the volume of the

dome at each time interval. The heights at each time interval can be modeled by a different

function, as the dome shape changes as time baking passes. These functions were determined

using Geogebra, where I uploaded each picture taken of the cakes baking at each time interval.

As displaying every calculation for each time interval would be redundant, I will only include a

sample calculation for one time interval (35 minutes, with an average height of 3.85 cm), where

the function 𝑓(𝑥) = −𝑥 2 represents the cake dome.

𝑏
𝐴=∫ 𝑓(𝑥)𝑑𝑥
𝑎
𝑏
𝑉=∫ 𝜋𝑓(𝑥)2 𝑑𝑥
𝑎
𝑏
𝑉 = 𝜋∫ 𝑓(𝑥)2 𝑑𝑥
𝑎
𝑏
𝑉 = 𝜋∫ (−𝑥 2 )2 𝑑𝑥
𝑎
 𝑏
𝑉 = 𝜋∫ 𝑥 4 𝑑𝑥
𝑎
𝑏
𝑥5
𝑉 = 𝜋∫
𝑥 𝑎
5 3.85
𝑥
𝑉 = 𝜋∫
𝑥 0
3.85
(3.85)5
𝑉= 𝜋∫
(3.85) 0

Reflection

Conclusion

Bibliography
Appendix