a person cleared a rectangular plot of land and covered it with gravel. then he got 9 square wooden sections of side lengths: 2,5,7,9,16,25,33 and 36. by placing the squares on the gravel with no two overlaping, the person built a patio which exactly covered the graveled surface. determine the perimeter of the persons new patio.
You seem to have 8, not 9 squares
sorry it is 8 squares
To determine the perimeter of the person's new patio, we need to find out the dimensions of the rectangular plot of land.
We know that the person covered the plot of land with gravel, and then placed square wooden sections without any overlap to form the patio. Since the wooden sections are squares, we can assume that the rectangular plot of land is also a square.
Let's check the dimensions of each wooden section and see if we can find any patterns:
- The wooden section with a side length of 2 covers an area of 2^2 = 4 square units.
- The wooden section with a side length of 5 covers an area of 5^2 = 25 square units.
- The wooden section with a side length of 7 covers an area of 7^2 = 49 square units.
- The wooden section with a side length of 9 covers an area of 9^2 = 81 square units.
- The wooden section with a side length of 16 covers an area of 16^2 = 256 square units.
- The wooden section with a side length of 25 covers an area of 25^2 = 625 square units.
- The wooden section with a side length of 33 covers an area of 33^2 = 1089 square units.
- The wooden section with a side length of 36 covers an area of 36^2 = 1296 square units.
From the given information, we can clearly see that the areas of the wooden sections are perfect squares. This indicates that the dimensions of the rectangular plot of land are also perfect squares.
Now, let's arrange these wooden sections in decreasing order of their side lengths: 36, 33, 25, 16, 9, 7, 5, 2.
Since the person placed these wooden sections on the gravel with no overlap, the total area covered by the patio would be equal to the sum of the areas of all these wooden sections.
36^2 + 33^2 + 25^2 + 16^2 + 9^2 + 7^2 + 5^2 + 2^2 = 1296 + 1089 + 625 + 256 + 81 + 49 + 25 + 4 = 3425
So, the total area covered by the patio is 3425 square units.
Since the rectangular plot of land is a square, the area can be expressed as (side length)^2.
Therefore, (side length)^2 = 3425
Taking the square root of both sides, we get:
side length = √3425 ≈ 58.59 (rounded to 2 decimal places)
The perimeter of the person's new patio is equal to four times the side length of the square.
perimeter = 4 * side length = 4 * 58.59 = 234.35 (rounded to 2 decimal places)
Therefore, the perimeter of the person's new patio is approximately 234.35 units.