To calculate how many pieces of size 345 by 290 millimeters can be cut from a sheet of plywood measuring 9 feet 7 inches by 4 feet 11 inches, we first need to convert the dimensions from feet to millimeters.
1 foot = 304.8 millimeters

So, a 9.7 by 4.11 foot sheet of plywood is equivalent to:

9.7 feet * 304.8 millimeters/foot = 2951.6 millimeters (length)
4.11 feet * 304.8 millimeters/foot = 1255.328 millimeters (width)

Now, let's see how many pieces of size 345 by 290 millimeters can fit into this sheet.

For the length:
2951.6 millimeters / 345 millimeters per piece ≈ 8.55 pieces

For the width:
1255.328 millimeters / 290 millimeters per piece ≈ 4.33 pieces

Since you can't have a fraction of a piece, you need to round down to the nearest whole number because you can't fit a partial piece:

Along the length: 8 pieces Along the width: 4 pieces

So, you can cut a total of 8 × 4 = 32
8 × 4 = 32 pieces of size 345 by 290 millimeters from a 9.7 by 4.11 foot sheet of plywood.