a rectangle of board measures 21 m by 70 cm. Circular plates are to be cut from the board of diameter 14 cm. how many plates and board is left

There is not an optimal solution to this, as far as I know.

If you consider the board to be invisible squares of 70cm x 70cm, 30 of them side by side, you can then look at packing circles in squares, and you have the squares then 5 diameters x 5 diameters. Now for squares, there are optimal solutions http://hydra.nat.uni-magdeburg.de/packing/csq/d3.html and you can see for each square N=25, and you have 30 squares, so 30x25.

This math of packing is a very challenging field, and of much commercial value. Box sizes are determined with this math.

For area, compute board area, subtract the areas of the circles.

To find out how many circular plates can be cut from the rectangle board, as well as how much board is left, we need to calculate the possible number of plates and the remaining area.

First, let's convert the dimensions of the rectangle board into the same unit. We'll convert the width, which is given in centimeters, into meters:
Width = 70 cm = 70/100 m = 0.7 m

Now, let's calculate the area of the rectangle board:
Area of rectangle = Length × Width
Area of rectangle = 21 m × 0.7 m
Area of rectangle = 14.7 m²

Next, we need to calculate the area of a circular plate:
Radius (r) = Diameter ÷ 2 = 14 cm ÷ 2 = 7 cm = 7/100 m = 0.07 m
Area of circle = π × r² = 3.14 × (0.07 m)² ≈ 0.0154 m² (rounded to four decimal places)

Now, we can find the number of plates that can be cut from the board:
Number of plates = Area of rectangle ÷ Area of circle
Number of plates = 14.7 m² ÷ 0.0154 m² ≈ 954.5455 (rounded to four decimal places)

Since we can't have a fraction of a plate, we need to round down this value. Therefore, you can cut 954 circular plates from the rectangular board.

To find out how much board is left, we subtract the area of all the circular plates from the total area of the rectangle board:
Remaining board area = Area of rectangle - (Number of plates × Area of circle)
Remaining board area = 14.7 m² - (954 × 0.0154 m²)
Remaining board area ≈ 14.7 m² - 14.7018 m² (rounded to four decimal places)
Remaining board area ≈ 0.0002 m²

So, there is approximately 0.0002 square meters of board left after cutting the circular plates.