Babylonian Problem - circa 1800 BC

An area, A, consisting of the sum of two squares is 1000. The side of one square is 10 less than 2/3 of the side of the other square. What are the sides of the squares?

let x = the side of the 1st square

let 2x/3-10 = the side of the 2nd square ("10 less than 2/3 of the side of the other square")

The area of a square is x^2. Therefore, if the sum of the areas = 1000, than
x^2 + (2x/3-10)^2 = 1000

foil the polynomial:

x^2 + (4x^2/9 - 40x/3 + 100) = 1000

You can simplify it and use the quadriatic equation to solve. The numbers do not seem very friendly though - is this a calculator problem? (If so, just use the graphing function to find solutions.)

Side of square 1 is 30

Side of square 2 is 10

peter is twice a old as paul was when peter was asold as paul now. the combined ages of peter and paul is 56 years. how old are peter and paul now?

To solve this problem, we need to set up equations based on the given information and then find the values of the sides of the squares. Let's break it down step by step:

1. Let's assume the side of one square is "x."
2. According to the problem, the side of the other square is "10 less than 2/3 of x." We can write this as (2/3)x - 10.
3. The area of a square is calculated by squaring its side, so the area of the first square is x^2, and the second square's area is ((2/3)x - 10)^2.
4. The problem states that the sum of the areas of the two squares is 1000, so we can set up the equation x^2 + ((2/3)x - 10)^2 = 1000.

Now, we can solve this equation to find the values of x, which will be the sides of the squares.

To solve this equation, follow these steps:

1. Expand ((2/3)x - 10)^2 using the formula (a - b)^2 = a^2 - 2ab + b^2.

The equation becomes:
x^2 + (4/9)x^2 - (40/3)x + 100 = 1000.

2. Combine like terms:
(13/9)x^2 - (40/3)x + 100 = 1000.

3. Subtract 100 from both sides of the equation:
(13/9)x^2 - (40/3)x = 900.

4. Multiply both sides by 9 to eliminate fractions:
13x^2 - 120x = 8100.

5. Rearrange the equation to be in standard quadratic form:
13x^2 - 120x - 8100 = 0.

Now, you can solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of x. Once you have the values of x, substitute them back into the expression (2/3)x - 10 to find the sides of the squares.