Multiple Choice

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$$24$$
Correct Answer
B$$32$$
C$$45$$
Correct Answer
D$$60$$

Solution

Let the sides rectangle be $$15\mathrm{k}$$ and $$8\mathrm{k}$$ and side of square be $$\mathrm{x}$$ then $$(15\mathrm{k}-2\mathrm{x})(8\mathrm{k}-2\mathrm{x})\mathrm{x}$$ is volume.

$$\Rightarrow \mathrm{v}=2(2\mathrm{x}^{3}-23\mathrm{k}\mathrm{x}^{2}+60\mathrm{k}^{2}\mathrm{x})$$

For maximum volume, the derivative of V w.r.t. x should be zero.

$$\Rightarrow \displaystyle \frac{\mathrm{d}\mathrm{v}}{\mathrm{d}\mathrm{x}}|_{x=5}=0$$

$$6\mathrm{x}^{2}-46\mathrm{k}\mathrm{x}+60\mathrm{k}^{2}|_{\mathrm{x}=5}=0$$

$$6\mathrm{k}^{2}-23\mathrm{k}+15=0$$

$$\mathrm{k}=3,\ \displaystyle \mathrm{k}=\frac{5}{6}$$ Only $$\mathrm{k}=3$$ is permissible.

So, the sides are $$45$$ and $$24$$.

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