Answer:
5/27 square meters
Explanation:
To evaluate the area of the rectangle, we need to multiply its length by its width.
Given that the length of the rectangle is 0.5555… meters and the width is 0.3333… meters, we can write the equation as:
Area = Length × Width
Let's calculate each part separately:
Length = 0.5555…
Since the length has a recurring decimal pattern of 5, we can write it as:
Length = 0.5555...
= 0.5 + 0.05 + 0.005 + 0.0005 + ...
= 5/10 + 5/100 + 5/1000 + 5/10000 + ...
We can see that this is an infinite geometric series with a common ratio (r) of 1/10 and a first term (a) of 5/10. The sum of an infinite geometric series is defined as:
Sum = a / (1 - r)
Substituting the values in, we have:
Length = (5/10) / (1 - 1/10)
Length = (5/10) / (9/10)
Length = 5/9
Now let's calculate the width in the same way:
Width = 0.3333…
Since the width has a recurring decimal pattern of 3, we can write it as:
Width = 0.3333...
= 0.3 + 0.03 + 0.003 + 0.0003 + ...
= 3/10 + 3/100 + 3/1000 + 3/10000 + ...
Again, we can see that this is an infinite geometric series with a common ratio of 1/10 and a first term of 3/10. Calculating the sum, we have:
Width = (3/10) / (1 - 1/10)
Width = (3/10) / (9/10)
Width = 3/9
Simplifying, we have:
Width = 1/3
Now, we can calculate the area by multiplying the length and width:
Area = Length × Width
Area = (5/9) × (1/3)
Area = 5/27
Therefore, the area of the rectangle is 5/27 square meters.