Math Help! Pre-Calc Question.

A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Express the sum of the areas of these squares, A as a function of the length of th cut, x.

Also, find the coordinates of the minimum and interpret the meaning in a sentence.

I just need the first part but if someone can help me with both, that would be greatly appreciated

Let's say that you cut at 3 inches. x = 3. Now one of the new pieces of wire is 3 inches. To get the length of the other piece, subtract x, 3, from the original length, which was 8 inches. The length of that piece of wire, thus, is 8 - x.

• length of piece one = x = 3 inches
• length of piece two = 8 - x = 8 - 3 = 5 inches

Since a square has four equal sides, you need to divide the length of each new piece by 4. So the length of each side of the first square is 3/4 of an inch. The length of each side of the second square is (x - 3)/4 or 5/4 of an inch. To find the area of a rectangle, take the length times the width. You have two squares; that means the length and the width are the same. Therefore, the area of a square is the length squared.

The area of the smaller square should be x^2. The area of the bigger square should be (8 - x)^2.

Thus, the sum of the areas of the two squares can be given by the function...

f(x) = x^2 + (8 - x)^2

or

A = x^2 + (8 - x)^2

To find the minimum of the function, first you have to rearrange it to find its vertex (the function is a parabola because of the x^2).

f(x) = x^2 + (8 - x)(8 - x) <--- rewrite

f(x) = x^2 + x^2 - 16x + 64 <--- use the FOIL method (First, Outer, Inner, Last)

f(x) = 2x^2 - 16x + 64 <--- combine like terms

f(x) = 2(x^2 - 8x) + 64 <--- factor out a 2

f(x) = 2(x^2 - 8x + 16) + 64 - 32 <--- complete the square by adding 32 and subtracting 32 (if you don't know what completing the square is, Google it)