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# 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?

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## Hints

• Hint 1
What are the dimensions of the box formed in terms of $$x$$?
• Hint 2
What is the function that defines the volume of the box?
• Hint 3
How 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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