ISSN: 3030-3931, Impac ac o : 7,241 Volume 10, issue 2, Noyab 2025
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THE ROLE OF MATHEMATICAL CONCEPTS IN REAL-WORLD PROBLEMS
Bekchano a Shakhnoza Khusanboy qizi
(mas e s deg ee holde in Ma hema ics, U gench S a e
Uni e si y named a e Abu Rayhan Be uni,
Email: [email p o ec ed],
Telephone: +998937582508)
Anno a ion: This a icle shows how ma hema ics applies o eal-li e p oblems. I includes
designing a sus ainable box using calculus, sol ing he S able Ma ching P oblem wi h
algo i hms, and inding a dam’s a ea using in eg als, highligh ing ma h’s ole in sol ing
p ac ical challenges.
Key wo ds: op imiza ion, s able ma ching, algo i hms, in eg al, geome y, sus ainabili y.
In oduc ion. This pape p esen s h ee eal-wo ld applica ions o ma hema ics ha
demons a e how abs ac concep s can sol e p ac ical p oblems. The i s case applies
calculus o design a sus ainable package using minimal ma e ial. The second explo es
compu a ional hinking h ough he S able Ma ching P oblem, showing how algo i hms can
ensu e ai ness in assignmen s. The hi d uses in eg al calculus o de e mine he a ea o a
g a i y dam’s c oss-sec ion, combining geome y and in eg a ion. Toge he , hese examples
highligh he e sa ili y o ma hema ical ools in op imizing design, suppo ing decision-
making, and sol ing enginee ing challenges e icien ly.
Ra ional unc ion o sus ainable package: a case s udy (Calculus)
Imagine a packaging company needs o design a shipping box wi h a squa e base and a
olume o
600
cubic inches. Wha dimensions should i ha e? Should he base be
5 5
inches,
10 10
inches, o some hing else? Which one uses he leas ma e ial? (
1
inch =
2.54
cm)
Solu ion:
We know he olume o he box mus be
600
cubic inches, so:
2
600 l h=
Now, subs i u e his in o he su ace a ea o mula:
2 2
2
600 2400
4 2 2S l l l
l l
= + = +
ISSN: 3030-3931, Impac ac o : 7,241 Volume 10, issue 2, Noyab 2025
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Le 's now ind he minimum su ace a ea using calculus by aking he de i a i e o he su ace
a ea unc ion:
( )
22400
2S l l l
= +
2
2
2400 2400
2 4
dS d l l
dl dl l l
= + = -
.
To ind c i ical poin s (possible minimums), we will sol e:
2
2400
4 0ll
- =
3600 8.43l= =
Second de i a i e es o check i i is a minimum
2
2 2 3
2400 4800
4 4
d S d l
dl dl l l
= - = +
Since his is always posi i e o
0l>
, he unc ion has a minimum a
8.43l=
. F om his
we can always ind
8.43h=
.
This means he box is a cube wi h all sides app oxima ely
8.43
inches, and i uses he leas
ma e ial o a ixed olume o 600 cubic inches.
The S able Ma ching P oblem (Compu a ional Thinking)
E e y yea , housands o s uden s g adua e om medical schools and begin hei ca ee s by
applying o esidency aining posi ions in hospi als. Each medical s uden has a pe sonal
anking o he hospi als whe e hey would p e e o wo k, and each hospi al has i s own
anking o s uden s based on academic me i , in e iews, and ins i u ional i .
S uden s: John, Alice, Ma ia, E han, Noo .
Hospi als: Hospi al A, Hospi al B, Hospi al C, Hospi al D, Hospi al E.
S uden
1s
2nd
3 d
4 h
5 h
John
B
A
D
E
C
Alice
A
C
B
D
E
Ma ia
B
D
E
A
C
ISSN: 3030-3931, Impac ac o : 7,241 Volume 10, issue 2, Noyab 2025
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E han
C
B
A
D
E
Noo
D
E
C
B
A
Hospi al
1s
2nd
3 d
4 h
5 h
A
E han
Alice
John
Ma ia
Noo
B
John
Ma ia
Alice
Noo
E han
C
E han
Noo
Alice
John
Ma ia
D
Noo
Ma ia
E han
Alice
John
E
Alice
E han
Noo
Ma ia
John
Solu ion. Le ’s sol e using Gale–Shapley.
S ep 1: E e yone p oposes o 1s choice
John → B
Alice → A
Ma ia → B
E han → C
Noo → D
Hospi als e alua e:B:John, Ma ia → p e e s John (1s ), ejec s Ma ia
A:Alice → accep s
C:E han → accep s
D:Noo → accep s
S ep 2: Ma ia p oposes o 2nd choice
D al eady holds Noo . D anks: Noo (1s ), Ma ia (2nd) → keeps Noo , ejec s Ma ia again.
S ep 3: Ma ia p oposes o 3nd choice: E is ee, accep s Ma ia.
Final ma ching:
S uden
Hospi al
John
B
Alice
A
Ma ia
E
ISSN: 3030-3931, Impac ac o : 7,241 Volume 10, issue 2, Noyab 2025
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E han
D
Noo
C
A ea o he G a i y Dam (Calculus)
Image 1.
You can see ha he g a i y dam's c oss-sec ion is composed o h ee undamen al geome ic
shapes: a ec angle, a igh iangle, an a ea unde he pa abola.
Image 2.
2
389 16 6224
A
A ea h b = = =
2
1 1 389 59 11475
2 2
B
A ea h a = = =
We can b eak i in o hese h ee pieces. We can use simple geome y o calcula e he a ea o
he ec angula piece and he iangula piece, which lea es us wi h calcula ing he a ea o he
pa abolic piece. This is whe e he calculus will be used.
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Image 3.
We a e using in eg al calculus o calcula e his. I we keep making he ec angles smalle and
smalle , we will e en ually ge he exac a ea unde he cu e. The cu e s a s a he o igin
( )
0,0
and ises o a heigh o
389
a he end o he
54
wide base.
To model he cu e, we use a simple quad a ic unc ion:
2
( )
C
x ax=
Using he known poin
( )
54,389
, we ind he alue o
a
:
389
2916
a=
.
So he unc ion becomes:
2
389
( ) 2916
C
x x=
.
Now we will use Riemann in eg al:
( ) ( )
0
lim i
b
C C i i C
xa
i
A ea x x x dx
®
= =
VV
( )
3
54 54 2 2
0 0
389 389 54 5803
2916 2916 3
C C
A ea x dx x dx = = = »
2
23502To alA ea =
.
ISSN: 3030-3931, Impac ac o : 7,241 Volume 10, issue 2, Noyab 2025
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Re e ences:
1. Kehle, P., & Ca pen e , T. (2016). Fai and S able Ma chings: Compu a ional Thinking
Module (S uden Edi ion). COMAP, Inc., in conjunc ion wi h DIMACS, Ru ge s Uni e si y.
2. Ra ional Func ion o Sus ainable Package: A Case S udy. Re ie ed om
h ps://da e .com/blog/ unc ions-sus ainabili y-1/
3. AP Calculus BC P ojec : The Dam P oblem – Cons uc ing a Dam ac oss he Ri e .
Re ie ed om h ps://www.ms.uky.edu/
4. Dams. Re ie ed om h ps://damsa e y.o g/
