Question

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are

• A. 24
• B. 32
• C. 45
• D. 60

Answer 1

Answer: (A), (C)

45

Let the sides of the rectangle be 15k and 8k and sides of square be x. Then $(15k-2x)(8k-2x)x$ is the volume.
$\therefore V=2\left(2x^3-23{\mathrm{kx}}^2+60k^{2}x\right)\because \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{squares}\mathrm{cut}\mathrm{is}100\mathrm{from}\mathrm{all}\mathrm{four}\mathrm{corners}.\therefore \mathrm{Area}\mathrm{of}\mathrm{each}\mathrm{square}\mathrm{is}25\mathrm{and}\mathrm{so}\mathrm{side}\mathrm{will}\mathrm{be}5.x=5.{\left(\frac{\mathrm{dV}}{\mathrm{dx}}\right)}_{x=5}=0\Rightarrow {\left(6x^2-46\mathrm{kx}+60k^2\right)}_{x=5}=0\Rightarrow 6k^2-23k+15=0\Rightarrow k=3\mathrm{or}\frac{5}{6}.\mathrm{Only}k=3\mathrm{is}\mathrm{permissible}.\mathrm{So},\mathrm{sides}\mathrm{are}45\mathrm{and}24.$