SOLUTION: a wire 10 cm long is cut into two pieces, one of length x and the other of 10 - x. each piece is bent into the shape of a square (a) find a function that models the total area enc

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Question 168001: a wire 10 cm long is cut into two pieces, one of length x and the other of 10 - x. each piece is bent into the shape of a square
(a) find a function that models the total area enclosed by the two squares.
(b) Find the Value of x that minimizes the total area of the two squares
Answer by gonzo(654) (Show Source):
You can put this solution on YOUR website!
area of a square is s^2 where s is the length of each side.
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the wire is cut into 2 lengths and forms a square with each.
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the wire of x length forms 4 sides with the length of each side equal to x/4.
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the wire of (10-x) length forms 4 sides with the length of each side equal to (10-x)/4.
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area of the square made by the wire of x length is
this equals which equals
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area of the square made by the wire of (10-x) length is .
this equals which equals .
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the total area enclosed by these squares is given by the equation:

since the denominator is equal, this equation becomes:

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since x^2 is positive, this is a quadratic equation that opens upward and points downward.
the point where the graph turns is a minimum point.
that point can be found as follows:
take the given formula and multiply it out to get a quadratic equation in standard form.
that standard form is:

that becomes:

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in that equation, the a coefficient is 1/8, and the b coefficient is -10/8.
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the formula gives the minimum/ maximum point of a quadratic equation.
substituting, that formula becomes:

when x = 5, y = 3.125
that's the minimum value of the equation.
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graph of the resulting equation looks like this showing that when x = 5, y = 3.125 and the value of the equation of the squares is at a minimum.