Use the above to make a conjecture about the area occupied by circles fitted into a rectangle Class 9

Use the above to make a conjecture about the area occupied by circles fitted into a rectangle Class 9

Question 1.

Use the above to make a conjecture about the area occupied by circles fitted into a rectangle in the manner shown. Test your conjecture for particular cases: 10 circles; 20 circles; 50 circles. Then prove your conjecture! Class 9

Solution:

Proof:

Conjecture: The fraction of a rectangle j covered by any number of identical circles ! arranged in a single row (touching the edges and each other) is always $\frac{\pi}{4}$.

Let number of circles = n, radius = r

From the figure,

∵ Height of rectangle = 2r

There are n circles in a row

∴ Length of rectangle = n × 2r = 2nr

Area of rectangle = Length × Breadth

= (2nr)(2r) = 4nr²

Area of circle = πr²

Area of n circles = nπr²

Fraction of rectangles covered by n circles

$=\frac{\text { Area of ' }^{\prime} n^{\prime} \text { Circles }}{\text { Area of Rectangle }}=\frac{n \pi r^2}{4 n r^2}$ ....(i)

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

∴ Fraction covered is always $\frac{\pi}{4}\left(\approx \frac{11}{14}\right)$ and is independent of number of circles

Verification:

For 10 circles:

Substituting n = 10 in (i), we get

Fraction of rectangles covered by 10 circles

$=\frac{10 \pi r^2}{40 r^2}=\frac{\pi}{4}$

For 20 circles:

Substituting n = 20 in (i), we get

Fraction of rectangles covered by 20 circles

$=\frac{20 \pi r^2}{80 r^2}=\frac{\pi}{4}$

For 50 circles:

Substituting n = 50 in (i), we get

Fraction of rectangles covered by 50 circles

$=\frac{50 \pi r^2}{200 r^2}=\frac{\pi}{4}$

∴ Fraction does not depend on n.

∴ Fraction of rectangle covered by circles = $\frac{\pi}{4}$