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IB Maths AI SL 5.7 Optimization

IB Maths AI SL 5.7 Optimization
IB Mathematics: applications and interpretation guide, first assessment 2021

Students practise applying calculus to real-world problems where a quantity must be maximised or minimised under constraints.

How this is tested

  • substitute constraint into surface area formula, differentiate, and solve dA/dw=0
  • interpret minimum point of profit model as optimal time to buy shares, reading t-coordinate from graph
  • recognise that hemisphere minimises surface area for given volume — deduced from optimisation result r=h

Question 13

[Maximum number: 5]

Soraya is designing a gift box in the shape of a rectangular prism, as shown in the diagram.

Figure for Question 13 — IB Maths AI SL

The length, l, of the box is 1.5 times the width, w, of the box. This can be expressed as l=1.5 w. The height, h, of the box is also labelled on the diagram.

The box has a volume of 10000 cm310000 \mathrm{~cm}^{3}. Soraya wants to minimize the surface area of the box. Find the width of the box.