Question : Nine times the area of a circle is the same as the three times the area of a square. What is the ratio of the diameter of the circle and the diagonal of the square?
Option 1: $\sqrt{2}: \sqrt{3 \pi}$
Option 2: $2: \sqrt{3 \pi}$
Option 3: $2: 3 \pi$
Option 4: $\sqrt{5}: \sqrt{7 \pi}$
Correct Answer: $\sqrt{2}: \sqrt{3 \pi}$
Solution :
Let the radius of the circle be $r$ and the side of the square be $a$.
According to the question,
⇒ $9 \times \pi r^2 = 3 \times a^2$
⇒ $a = r \sqrt{3\pi}$
The diameter, $d$ of the circle = $2r$
The diagonal, $D$ of the square = $a\sqrt{2}$
Putting the value of $a$, we get:
$D = a\sqrt{2} = r\sqrt{6\pi}$
Therefore, the ratio of the diameter of the circle to the diagonal of the square is:
$\frac{d}{D} = \frac{2r}{r\sqrt{6\pi}} = \frac{\sqrt2}{\sqrt{3\pi}}$
Hence, the correct answer is $\sqrt{2}: \sqrt{3 \pi}$.
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