# Exam-Style Question on Differentiation Optimisation

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 110. This question is similar to one that appeared on an IB Studies paper in 2014. The use of a calculator is allowed.

A package is in the shape of a cuboid and has a length $$l$$ cm, width $$w$$ cm and height of 12 cm.

(a) Express the volume of the package in terms of $$l$$ and $$w$$.

The total volume of the package is 2400 cm3.

(b) Show that $$l=\frac{200}{w}$$.

The package is tied up using a length of red string that fits exactly around the package in two different directions, as shown in the following diagram (not to scale).

(c) Show that the length of string, $$x$$cm, required to tie up the package can be written as $$24+4w+\frac{400}{w}$$

(d) Sketch the graph of $$x$$ for $$0\lt w \le 12$$, clearly showing the local minimum point.

(e) Find $$\frac{dx}{dw}$$.

(f) Find the value of $$w$$ for which $$x$$ is a minimum.

(g) Find the value, $$l$$, of the package for which the length of string is a minimum.

(h) Find the minimum length of string required to tie up the package.

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