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TIFR-2012-Maths-D: 34

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True/False Question:

If a rectangle $R:=\left \{ \left ( x,y \right ) \in \mathbb{R}^{2}\mid A\leq x\leq B,C\leq y\leq D\right \}$ can be covered (allowing overlaps ) by $25$ discs of radius $1$ then it can also be covered by $101$ dics of radius $\frac{1}{2}.$

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$\text{Area of 25 circles of radius 1} \geq \text{Area of rectangle}$

$25 * \pi * (1)^2 \geq (B – A) * (C – D)$

$25 * \pi \geq (B – A) * (C – D)$

Now $\text{Area of 101 circles of radius } \frac{1}{2} = 25 * \pi * (\frac{1}{2})^2 = 25.25\pi$

$\therefore \text{Area of 101 circles of radius } \frac{1}{2} \geq \text{Area of rectangle}$

So True.
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