A circle of largest area is cut from a rectangular piece of card-board with dimensions 55 cm and 42 cm. Find the ratio between the area of the circle cut and the area of the remaining card-board.

Given:

The dimensions of rectangular piece of card-board are:

Length = 55 cm

Width = 42 cm

The largest circle that can be cut from the rectangle will have a diameter equal to the shorter side of the rectangle.

Diameter = Width = 42 cm

∵ Radius = r = $\dfrac{d}{2}$ = $\dfrac{42}{2}$ = 21 cm

Area of the circle = πr²

$= \dfrac{22}{7} \times 21^2\\[1em] = \dfrac{22}{7} \times 441\\[1em] = \dfrac{9,704}{7}\\[1em] = 1,386 \text{ cm}^2$

Area of the rectangular piece of cardboard = 55 x 42 cm²

= 2,310 cm²

Therefore, area of remaining cardboard = Area of rectangle - Area of circle

= (2,310 - 1,386) cm²

= 924 cm²

So, the ratio between the area of the circle cut and the area of the remaining card-board = 1,386 : 924

= 231 : 154

= 3 : 2

Hence, the ratio between the area of the circle cut and the area of the remaining cardboard is 3 : 2.