Answer:

Ancient Word Problem 1:

Method Used by Babylonians

Problem: In ancient Babylon, a farmer had a square piece of land. He wanted to increase the area of his land by 60 units, while keeping the length of each side an integer value. What were the dimensions of the original square piece of land?

Solution:

The Babylonians used a method called the "completing the square" method. They would take the original area, add the given increase, and then find the square root of the result.

Let's represent the side length of the original square as x. The area of the original square is x^2. The farmer wants to increase the area by 60 units, so the new area is x^2 + 60.

By using the "completing the square" method, we can solve for x using the formula (x^2 + 60) = (x + (60/(2x)))^2.

Using algebra and Babylonian methods of calculation, we can find that the original side length x is 6. Therefore, the dimensions of the original square piece of land were 6 units by 6 units.

Ancient Word Problem 2:

Thomas Carlyle's Geometric Solution

Problem: In ancient Greece, a farmer wanted to enclose a rectangular piece of land with a perimeter of 28 units. If the length of the land is 4 units longer than the width, what are the dimensions of the land?

Solution:

We can use Thomas Carlyle's geometric solution to solve this problem by forming an equation for the perimeter of the rectangle.

Let's represent the width of the land as x. Since the length of the land is 4 units longer than the width, the length is (x + 4). The perimeter of the rectangle is given by P = 2x + 2(x + 4) = 28.

Using this perimeter equation, we can solve for the width x by rearranging the equation. After finding the width, we can then calculate the length by adding 4 units to the width.

Ancient Word Problem 3:

Pythagorean Geometry Solution

Problem: Pythagoras has a square piece of land with an area of 100 square units. He wants to build a rectangular fence around the land, where the width of the fence is half the length of its diagonal. What are the dimensions of the rectangular fence?

Solution:

For the Pythagorean geometry solution, we can use the Pythagorean theorem to calculate the diagonal and then find the dimensions of the rectangular fence.

The area of the original square land is 100 square units, so the side length of the square is 10 units. Using the Pythagorean theorem, we can calculate the diagonal of the square, which is 10√2 units.

Since the width of the fence is half the length of the diagonal, the width is 5√2 units. The length of the rectangular fence can be calculated using the given width.

I hope these solutions demonstrate the application of different ancient methods to solve word problems!

Step-by-step explanation:

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