SOLUTION: A piece of wire 6 metres long is cut into two parts, one of which is used to form a square and the other to form a rectangle whose length is three times its width. If x is the leng

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Question 1123737: A piece of wire 6 metres long is cut into two parts, one of which is used to form a square and the other to form a rectangle whose length is three times its width. If x is the length of the wire used to form the square
A) find an expression for the area of the square
B) an expression for the area of the rectangle
C) the lengths of the two parts of the sum of the areas is a minimum
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A piece of wire 6 metres long is cut into two parts, one of which is used to form a square and the other to form a rectangle whose length is three times its width.
:
If x is the length of the wire used to form the square:
then
(6-x) = the length to form the rectangle
:
A) find an expression for the area of the square.
A(x) = %28x%2F4%29%5E2
B) an expression for the area of the rectangle
let w = the width of the rectangle
then
3w = the width
and
3w%5E2 = the area
the relationship of w to (6-x); the perimeter of the rectangle
2w + 6w = (6-x)
8w = (6-x)
w = %286-x%29%2F8
find the area in terms of x, (A = 3w%5E2) replace with x
A(x) = 3%28%286-x%29%2F8%29%5E2 is the area of the rectangle
:
C) the lengths of the two parts of the sum of the areas is a minimum
A(x) = %28x%2F4%29%5E2 + 3%28%286-x%29%2F8%29%5E2
the easiest way to find the minimum area is to graph it, y = the area

minimum area when x = 2.5 m the length to form the square
and
6 - 2.5 = 3.5 m the length to form the rectangle