Question 18

We will assume circles are packed in rows and columns, touching each other.

Step 1 — Make a conjecture

Let's consider a rectangle filled with circles. Each circle has a radius of $r$. The area of one circle is $\pi r^2$. Let there be $N$ circles in total. The total area of all circles is $N \times \pi r^2$. Let the rectangle have $m$ rows and $n$ columns of circles. So, $N = m \times n$. The width of the rectangle is $n$ times the diameter of a circle. The diameter of a circle is $2r$. The width of the rectangle is $n \times 2r$. The height of the rectangle is $m$ times the diameter of a circle. The height of the rectangle is $m \times 2r$. The area of the rectangle is width times height.

$\text{Area of rectangle} = (n \times 2r) \times (m \times 2r)$

$= n \times m \times 4r^2$

$= N \times 4r^2$

Now, let's find the ratio of circle area to rectangle area.

$\text{Ratio} = \frac{\text{Total area of circles}}{\text{Area of rectangle}}$

$= \frac{N \times \pi r^2}{N \times 4r^2}$

$= \frac{\pi}{4}$

$\boxed{\text{The area occupied by circles is } \frac{\pi}{4} \text{ of the rectangle's area.}}$

Step 2 — Test for 10 circles

Let's arrange 10 circles in a $2 \times 5$ grid. Each circle has a radius of 1 unit. Number of rows $m = \mathbf{2}$. Number of columns $n = \mathbf{5}$. The rectangle width is $5 \times (2 \times 1) = \mathbf{10}$ units. The rectangle height is $2 \times (2 \times 1) = \mathbf{4}$ units. The area of the rectangle is $10 \times 4 = \mathbf{40}$ square units. The area of one circle is $\pi \times (1)^2 = \pi$ square units. The total area of 10 circles is $10 \times \pi = \mathbf{10\pi}$ square units. Let's check our conjecture: $\frac{\pi}{4} \times \text{Area of rectangle}$.

$\frac{\pi}{4} \times 40$

$= 10\pi$

$\boxed{\text{The conjecture holds for 10 circles.}}$

Step 3 — Test for 20 circles

Let's arrange 20 circles in a $4 \times 5$ grid. Each circle has a radius of 1 unit. Number of rows $m = \mathbf{4}$. Number of columns $n = \mathbf{5}$. The rectangle width is $5 \times (2 \times 1) = \mathbf{10}$ units. The rectangle height is $4 \times (2 \times 1) = \mathbf{8}$ units. The area of the rectangle is $10 \times 8 = \mathbf{80}$ square units. The total area of 20 circles is $20 \times \pi = \mathbf{20\pi}$ square units. Let's check our conjecture: $\frac{\pi}{4} \times \text{Area of rectangle}$.

$\frac{\pi}{4} \times 80$

$= 20\pi$

$\boxed{\text{The conjecture holds for 20 circles.}}$

Step 4 — Test for 50 circles

Let's arrange 50 circles in a $5 \times 10$ grid. Each circle has a radius of 1 unit. Number of rows $m = \mathbf{5}$. Number of columns $n = \mathbf{10}$. The rectangle width is $10 \times (2 \times 1) = \mathbf{20}$ units. The rectangle height is $5 \times (2 \times 1) = \mathbf{10}$ units. The area of the rectangle is $20 \times 10 = \mathbf{200}$ square units. The total area of 50 circles is $50 \times \pi = \mathbf{50\pi}$ square units. Let's check our conjecture: $\frac{\pi}{4} \times \text{Area of rectangle}$.

$\frac{\pi}{4} \times 200$

$= 50\pi$

$\boxed{\text{The conjecture holds for 50 circles.}}$

Step 5 — Prove the conjecture

Let's consider a rectangle. It contains $N$ identical circles. Each circle has a radius $r$. The circles are arranged in $m$ rows. They are also arranged in $n$ columns. The total number of circles $N = m \times n$. The area of one circle is $\pi r^2$. The total area occupied by all circles is $N \pi r^2$. The diameter of each circle is $2r$. The width of the rectangle is $n$ times the diameter.

$\text{Width of rectangle} = n \times (2r)$

The height of the rectangle is $m$ times the diameter.

$\text{Height of rectangle} = m \times (2r)$

The area of the rectangle is width times height.

$\text{Area of rectangle} = (n \times 2r) \times (m \times 2r)$

$= n \times m \times 4r^2$

$= N \times 4r^2$

Now, we find the ratio of the areas.

$\frac{\text{Total area of circles}}{\text{Area of rectangle}} = \frac{N \pi r^2}{N 4r^2}$

$= \frac{\pi}{4}$

This ratio is always $\frac{\pi}{4}$. The area occupied by circles is $\frac{\pi}{4}$ of the rectangle's area.

Answer

(i) The conjecture is that the area occupied by the circles is $\frac{\pi}{4}$ times the area of the rectangle. (ii) The conjecture holds for 10, 20, and 50 circles. (iii) The proof shows the ratio of the total circle area to the rectangle area is always $\frac{\pi}{4}$.