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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What are the dimensions of the box formed in terms of
$x$? -
What is the function that defines the volume of the box?
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How may we find the maximum point of that function?
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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$V\left(\frac{5}{3}\right)=\frac{2000}{27}.$