Alice purchased 30 decorative stone square blocks to create an enclosed rectangular garden area. Each square block measures 1 foot by 1 foot, and one possible arrangement is shown. Including the arrangement shown, in how many distinct ways can Alice arrange the blocks to enclose a garden, and for which arrangement is the garden area greatest?
To find the number of distinct ways Alice can arrange the blocks to enclose a garden, we need to find the number of distinct rectangles that can be formed using these blocks.
Now, let's break down the problem step-by-step:
Step 1: Calculate the number of possible lengths and widths for the garden using the 30 blocks.
Since each square block measures 1 foot by 1 foot, we need to find the factors of 30.
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.
Step 2: Determine the number of possible rectangles.
To find the number of possible rectangles, we need to count the number of unique combinations of lengths and widths.
For each factor of 30, there is a corresponding unique length and width combination.
For example, if the length is 1 foot, then the width is 30 feet, and vice versa.
So, we have 8 unique combinations.
Therefore, there are 8 distinct ways Alice can arrange the blocks to enclose a garden.
Step 3: Find the arrangement that results in the greatest garden area.
To find the arrangement with the greatest garden area, we need to determine the dimensions of each possible rectangle and calculate the corresponding area.
The rectangles are:
1 x 30 = 30 square feet
2 x 15 = 30 square feet
3 x 10 = 30 square feet
5 x 6 = 30 square feet
6 x 5 = 30 square feet
10 x 3 = 30 square feet
15 x 2 = 30 square feet
30 x 1 = 30 square feet
Therefore, for all 8 arrangements, the garden area is 30 square feet. There is no arrangement with a greater area since all the rectangles have the same area.
In summary:
- There are 8 distinct ways Alice can arrange the blocks to enclose a garden.
- The garden area is the greatest for all arrangements, having an area of 30 square feet.
To find the number of distinct ways Alice can arrange the blocks to enclose a garden, we can use combinatorics.
First, let's consider the dimensions of the rectangular garden area. Since each square block measures 1 foot by 1 foot, the length and width of the garden area must be positive integers.
The total number of ways to arrange the blocks can be found by considering the divisors of 30. The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. For each divisor, we can determine the corresponding length and width of the garden area.
Let's go through the divisors:
1: If the length is 1, the width must be 30. If the width is 1, the length must be 30. So there are 2 arrangements for this divisor.
2: If the length is 2, the width must be 15. If the width is 2, the length must be 15. Again, there are 2 arrangements.
3: If the length is 3, the width must be 10. If the width is 3, the length must be 10. Once again, 2 arrangements.
5: If the length is 5, the width must be 6. If the width is 5, the length must be 6. 2 arrangements.
6: If the length is 6, the width must be 5. If the width is 6, the length must be 5. 2 arrangements.
10: If the length is 10, the width must be 3. If the width is 10, the length must be 3. 2 arrangements.
15: If the length is 15, the width must be 2. If the width is 15, the length must be 2. 2 arrangements.
30: If the length is 30, the width must be 1. If the width is 30, the length must be 1. 2 arrangements.
Now let's sum up the total number of arrangements:
2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16
Therefore, there are 16 distinct ways Alice can arrange the blocks to enclose a garden.
To find the arrangement where the garden area is greatest, we need to find the pair of divisors that multiplies to 30 and gives the largest possible product.
Since 30 = 2 * 3 * 5, the longest side of the garden will be 30 and the other two sides will be 1. Therefore, the garden area is greatest when the arrangement is 30 by 1.
Thus, the arrangement with the greatest garden area is a rectangular garden with dimensions 30 feet by 1 foot.
6x5 gives greatest area
Just think of the factors of 30, and list them in pairs for the others.