Math Exploration – Higher Level
Topic: To determine the optimal ratio of length, width and height of an open bookshelf
having n levels in order to maximize its volume.
Aim
The aim of this exploration is to determine the dimensions of an open book shelf with "n"
levels that would maximise its volume given a given surface area of a wooden ply.
Methodology
I will begin by figuring out the expression for the open book shelf's volume and surface
area, which has the shape of a cuboid. For a given surface area, the volume must be
maximised. Therefore, the constraint in this optimisation issue is surface area. The
formula to determine the ideal dimensions given surface area ‘S’ and ‘n’ number of
levels will be established using multivariable calculus and the Lagrange multiplier
technique, as I realised while working on the subject that single variable calculus would
not be appropriate. Furthermore, I'll be calculating the volume for cabinets having different
number of levels to ascertain their relationship.
Mathematical Working
The bookshelf is cut in such a way that it appears as below –
1
, Since we are designing the cabinet having “𝑛” levels and having dimensions as length 𝑎
metres, width 𝑏 metres and height as 𝑐 metres, the area 𝑎𝑏 would be added for each of the
𝑛 levels since (𝑛 ∈ 𝑁).
The volume of the cabinet 𝑉 :
𝑉 = 𝑎𝑏𝑐
Surface Area of the cabinet 𝑆𝑐 :
𝑆𝑐 = (2𝑎𝑏) + (2𝑏𝑐) + (𝑎𝑐) + (𝑛 × 𝑎𝑏) = (𝑛 + 2)𝑎𝑏 + 𝑎𝑐 + 2𝑏𝑐
The above expression has 3 variables involved. Since direct algebraic reduction to a single
variable while satisfying the constraint is not optimal, we will use the multivariable calculus.
We will explore the partial derivative concept below.
Lagrange Multipliers
This method helps in the optimisation use cases, where we frequently seek to maximise an
objective function, such as production or profit, while taking financial or resource constraints
into consideration. The primary idea is to enter these limitations into the objective functions,
by having additional variables, called Lagrange multipliers.
2
Topic: To determine the optimal ratio of length, width and height of an open bookshelf
having n levels in order to maximize its volume.
Aim
The aim of this exploration is to determine the dimensions of an open book shelf with "n"
levels that would maximise its volume given a given surface area of a wooden ply.
Methodology
I will begin by figuring out the expression for the open book shelf's volume and surface
area, which has the shape of a cuboid. For a given surface area, the volume must be
maximised. Therefore, the constraint in this optimisation issue is surface area. The
formula to determine the ideal dimensions given surface area ‘S’ and ‘n’ number of
levels will be established using multivariable calculus and the Lagrange multiplier
technique, as I realised while working on the subject that single variable calculus would
not be appropriate. Furthermore, I'll be calculating the volume for cabinets having different
number of levels to ascertain their relationship.
Mathematical Working
The bookshelf is cut in such a way that it appears as below –
1
, Since we are designing the cabinet having “𝑛” levels and having dimensions as length 𝑎
metres, width 𝑏 metres and height as 𝑐 metres, the area 𝑎𝑏 would be added for each of the
𝑛 levels since (𝑛 ∈ 𝑁).
The volume of the cabinet 𝑉 :
𝑉 = 𝑎𝑏𝑐
Surface Area of the cabinet 𝑆𝑐 :
𝑆𝑐 = (2𝑎𝑏) + (2𝑏𝑐) + (𝑎𝑐) + (𝑛 × 𝑎𝑏) = (𝑛 + 2)𝑎𝑏 + 𝑎𝑐 + 2𝑏𝑐
The above expression has 3 variables involved. Since direct algebraic reduction to a single
variable while satisfying the constraint is not optimal, we will use the multivariable calculus.
We will explore the partial derivative concept below.
Lagrange Multipliers
This method helps in the optimisation use cases, where we frequently seek to maximise an
objective function, such as production or profit, while taking financial or resource constraints
into consideration. The primary idea is to enter these limitations into the objective functions,
by having additional variables, called Lagrange multipliers.
2
