Suppose that a wire 10 cm long is cut into two pieces, one of length x and the other of length 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.
a.) If A1=x2 be the area of the square with length x, then A2=(10−x)2 is the area of the other square. Thus, the total area is AT=A1+A2.
AT=x2+(10−x)2AT=x2+100−20x+x2AT=2x2−20x+100
b.) The function AT is a quadratic function with a=2 and b=−20, thus, its minimum value occurs when
x=−b2a=−(−20)2(2)=204=5 cm
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