The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field. 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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