Consider every whole number area from 2 square units to 30 square units. For which of these areas is there only one rectangle with whole number side lengths?
To find the rectangle with whole number side lengths for each area from 2 square units to 30 square units, we can systematically go through every possible rectangle and determine if its area matches the desired range.
First, let's list down all the possible areas from 2 to 30:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30
Now, we can proceed to find the rectangles with whole number side lengths for each of these areas by considering all possible combinations of length and width.
For each area, we will calculate the factors (whole number divisors) of that number. If there are exactly two factors, then it means there is only one rectangle possible with those dimensions.
Let's go through the process step by step:
1. Area 2:
Factors of 2: 1, 2
Since there are only two factors, there is only one rectangle with whole number side lengths for an area of 2.
2. Area 3:
Factors of 3: 1, 3
Once again, there are only two factors, so there is only one rectangle with whole number side lengths for an area of 3.
3. Area 4:
Factors of 4: 1, 2, 4
In this case, there are three factors, which means multiple rectangles are possible for an area of 4.
4. Continue this process for all the remaining areas.
Using this method, we can determine that the following areas have only one rectangle with whole number side lengths: 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 19, 21, 22, 26, 29.
Therefore, these are the areas for which there is only one rectangle with whole number side lengths in the given range (2 square units to 30 square units).