A piece of wire 24 cm long is cut into 2 pieces. Each piece is then formed into a square. If the sum of the areas of the 2 squares is 20 cm^2,...

Question

...(continue of the question) find the lengths of the twp pieces of wire.

Solve and type an equation

Answer 1

I am going to show you how to set up this difficult problem. First, call the two pieces something, like x and y. x + y = 24. Unfortunately, the two pieces are not of equal size. To make a square, we divide x and y by 4 each to get the sides.

So side1 = x/4 and side2 = y/4. And, we know that (side1)² + (side2)² = 20. Put in x/4 and y/4 in for the sides squared. Don't forget to square the 4.  You have x²/4² + y²/4⁴ = 20. Multiply through. You get x² + y² = 20x16 = 320.

 

Now you have to substitute and get a quadratic. Let's pick x. x = 24 - y from the first equation. We have to square this.

 

(24 - y)² + y² = 320. Foil the first part. You get 24² -2x24y + y² + y² = 320.

Simplify this to 2y² - 2x24y - 320 + 24² = 0   Solve this equation by multiplying the terms out. Then make it easier on yourself by dividing the result by 2, since 0/2 is still zero. You will have a quadratic and find the two zeros of this quadratic. Remember that the two zeros multiply to the last term and add to the middle term (24 times 2). Your result will be the two wire piece lengths. Verify that the areas of squares from these add up to 20. Done!

Answer 2

Intuitive answer: length # 1 = 8cm forming a 2 by 2 square with area of 4

 

length # 2 = 16 cm forming a 4 by 4 square with area of 16

 

You can (should) solve this with simultaneous equations: