Question

A wire 24 inches long is to be cut into 4 pieces to form a rectangle whose shortest side has a length of x.
a)Express the area A of the rectangle as a function of x.

Answer 1

12x -x^2

Answer 2

\[x(12-x)\] i think should work

Answer 3

the answer was a minute late, that should be in picoseconds!

Answer 4

so both would work??

Answer 5

idea is this
if x is one side and y the other then the perimeter is
\[2x+2y=24\] so
\[x+y=12\] and
\[y=12-x\] making the area (length times width)
\[A(x)=x(12-x)\]

Answer 6

yes they are the same
\[A(x)=x(12-x)=12x-x^2\]

Answer 7

ok and it says that i should make a conjecture about the dimensions that yeild a maximum area

Answer 8

make a square (more or less obvious)

Answer 9

i mean it is really obvious, because this is symmetric in x and y so it is biggest when they are equal. need some algebra to prove it?

Answer 10

sure

Answer 11

\[A(x)=12x-x^2\] a parabola that opens down. biggest at the vertex which is at
\[-\frac{b}{2a}=-\frac{12}{-2\times-1}=6\] so x should be 6 and therefore y should also be 6 and so it is a square with area = 6^2=36

Answer 12

hope you were not supposed to use calc or something

Answer 13

yea this is the first few pages of calc

Answer 14

what would be the domain?

Answer 15

well you could do the following ridiculous thing
1)
\[A(x)=12x-x^2\]

2)
\[A'(x)=12-2x\] set
\[12-2x=0\] get
\[x=6\] but it is alway
\[-\frac{b}{2a}\] for a quadratic

Answer 16

the domain if we were to graph that. because it asks us to graph also

Answer 17

well you are making a rectangle right? so negative lengths wouldn't make sense

Answer 18

and clearly x cannot be bigger than 24 because that is all you got

Answer 19

ok......

Answer 20

actually scratch that

Answer 21

x is only one side, so it cannot be bigger than 12

Answer 22

alright

Answer 23

i would say domain is

\[0<x<12\] and here is a nice picture
http://www.wolframalpha.com/input/?i=y%3D12x-x^2+for+0%3Cx%3C12

Answer 24

there now you must be done with this homework problem because we have beaten it to death

Answer 25

lol yes thanks you were very helpful XD