This problem involves calculating the number of tiles that can be made from a larger block of marble, considering the dimensions of both the block and the tiles, and accounting for material wastage during the cutting process.

Marble Block Volume Calculation

First, we determine the total volume of the marble block. The dimensions of the block are given as 5 m x 4 m x 2 m.

The volume of a rectangular prism (like the marble block) is calculated using the formula:

$$ V_{\text{block}} = \text{Length} \times \text{Width} \times \text{Height} $$

Plugging in the dimensions:

$$ V_{\text{block}} = 5 \text{ m} \times 4 \text{ m} \times 2 \text{ m} $$

$$ V_{\text{block}} = 40 \text{ m}^3 $$

So, the total volume of the marble block is 40 cubic meters.

Single Tile Volume Calculation

Next, we calculate the volume of a single rectangular tile. The dimensions of each tile are 1 m x 0.5 m with a thickness of 10 cm.

It is crucial to ensure all units are consistent. We convert the thickness from centimeters to meters:

Thickness = 10 cm = $ \frac{10}{100} $ m = 0.1 m

The volume of a single tile is calculated as:

$$ V_{\text{tile}} = \text{Length} \times \text{Width} \times \text{Thickness} $$

Plugging in the tile dimensions:

$$ V_{\text{tile}} = 1 \text{ m} \times 0.5 \text{ m} \times 0.1 \text{ m} $$

$$ V_{\text{tile}} = 0.05 \text{ m}^3 $$

Thus, the volume of one tile is 0.05 cubic meters.

Theoretical Number of Tiles

To find out how many tiles could theoretically be made if there were no wastage, we divide the total volume of the marble block by the volume of a single tile.

$$ \text{Theoretical Number of Tiles} = \frac{V_{\text{block}}}{V_{\text{tile}}} $$

$$ \text{Theoretical Number of Tiles} = \frac{40 \text{ m}^3}{0.05 \text{ m}^3} $$

$$ \text{Theoretical Number of Tiles} = \frac{40}{5/100} = 40 \times \frac{100}{5} = 40 \times 20 $$

$$ \text{Theoretical Number of Tiles} = 800 $$

Without considering wastage, 800 tiles could be made.

Accounting for Wastage

The problem states that there is a 10% wastage in cutting. This means that only 90% of the marble block's volume is effectively used to create usable tiles.

To find the actual number of tiles, we multiply the theoretical number of tiles by the percentage of usable material (100% - 10% = 90%).

$$ \text{Actual Number of Tiles} = \text{Theoretical Number of Tiles} \times (1 - \text{Wastage Percentage}) $$

$$ \text{Actual Number of Tiles} = 800 \times (1 - 0.10) $$

$$ \text{Actual Number of Tiles} = 800 \times 0.90 $$

$$ \text{Actual Number of Tiles} = 720 $$

Therefore, 720 tiles will be made after accounting for 10% wastage.