# Practice Paper 1 Question 1

You have a card of 10cm by 10cm. What is the largest volume in cm\(^3\) of a box (without a lid) that can be obtained by cutting out a square of side \(x\) from each corner and then folding the flaps up?

## Related topics

## Hints

- Hint 1What are the dimensions of the box formed in terms of \(x\)?
- Hint 2What is the function that defines the volume of the box?
- Hint 3How may we find the maximum point of that function?

## Solution

Cutting sides of length \(x\) and folding up the flaps results in a prism of height \(x\) and a square base of side \(10-2x\). The volume is therefore given by \(V(x)=x(10-2x)^2\). Differentiate this to find the stationary points:

\[V'(x)=(10-2x)^2-4x(10-2x)=4(x-5)(3x-5).\]

So \(V'(x)=0\) at \(x=5\) and \(x=\frac{5}{3}\). Substitute back in to find which of them gives the higher value (that's quicker than doing second derivatives), to get that \(\frac{5}{3}\) gives the maximum. Hence our final answer is \(V\left(\frac{5}{3}\right)=\frac{2000}{27}.\)

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