Sure, I will demonstrate the steps to solve this problem in detail.
Step 1: Understand and Convert Units
- The side of each tile is given as 40 cm.
- The perimeter of the circular floor is given as 66 meters.
Since we will eventually need to calculate areas in square centimeters, it is important to note that 1 meter = 100 centimeters. Therefore, the perimeter of the floor in centimeters is:
[tex]\[ 66 \, \text{m} = 66 \times 100 \, \text{cm} = 6600 \, \text{cm} \][/tex]
Step 2: Calculate the Radius of the Circle
The formula for the perimeter (circumference) of a circle is:
[tex]\[ \text{Perimeter} = 2 \pi r \][/tex]
Given that [tex]\(\pi = \frac{22}{7}\)[/tex] and the perimeter is 6600 cm, we rearrange the formula to solve for the radius [tex]\(r\)[/tex]:
[tex]\[ 6600 = 2 \times \frac{22}{7} \times r \][/tex]
[tex]\[ r = \frac{6600 \times 7}{2 \times 22} \][/tex]
[tex]\[ r = \frac{46200}{44} \][/tex]
[tex]\[ r = 1050 \, \text{cm} \][/tex]
Step 3: Calculate the Area of the Circular Floor
The formula for the area of a circle is:
[tex]\[ \text{Area} = \pi r^2 \][/tex]
Substituting the values of [tex]\( \pi \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ \text{Area} = \frac{22}{7} \times (1050)^2 \][/tex]
[tex]\[ \text{Area} = \frac{22}{7} \times 1102500 \][/tex]
[tex]\[ \text{Area} = \frac{22 \times 1102500}{7} \][/tex]
[tex]\[ \text{Area} = \frac{24255000}{7} \][/tex]
[tex]\[ \text{Area} = 3465000 \, \text{cm}^2 \][/tex]
Step 4: Calculate the Area of One Tile
The side length of each tile is given as 40 cm. Therefore, the area of one tile is:
[tex]\[ \text{Area of one tile} = 40 \, \text{cm} \times 40 \, \text{cm} = 1600 \, \text{cm}^2 \][/tex]
Step 5: Calculate the Number of Tiles Required
To find the number of tiles required to cover the entire floor, we divide the total area of the floor by the area of one tile:
[tex]\[ \text{Number of tiles} = \frac{\text{Total floor area}}{\text{Area of one tile}} \][/tex]
[tex]\[ \text{Number of tiles} = \frac{3465000}{1600} \][/tex]
[tex]\[ \text{Number of tiles} \approx 2165.625 \][/tex]
Since we cannot have a fraction of a tile, we round this up to the nearest whole number:
[tex]\[ \text{Number of tiles required} = 2166 \][/tex]
Therefore, the number of tiles required to completely tile the floor is 2166.
