Two posts, one 12 feet high and the other 22 feet high, stand
30 feet apart. They are held in place by two wires, attached
to a single stake, running from ground level to the top of
each post. If x represents the distance from the base of the
12-foot post to the stake, find an expression that
represents the total length of the wire in terms of x?
30 feet apart. They are held in place by two wires, attached
to a single stake, running from ground level to the top of
each post. If x represents the distance from the base of the
12-foot post to the stake, find an expression that
represents the total length of the wire in terms of x?
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If you draw a diagram the method of solution is pretty obvious. You have two right triangles, and the length of the wire is the sum of the lengths of their hypotenuses.
One triangle has legs of length 12 feet and x feet, so the hypotenuse of this triangle has length √(12² + x²) = √(144 + x²) feet.
The other triangle has legs of length 22 feet and (30 - x) feet, so the hypotenuse of this triangle has length √(22² + (30 - x)²) = √(484 + (900 - 60x + x²)) = √(1384 - 60x + x²) feet.
Thus, the length of the wire is √(144 + x²) + √(1384 - 60x + x²) feet.
One triangle has legs of length 12 feet and x feet, so the hypotenuse of this triangle has length √(12² + x²) = √(144 + x²) feet.
The other triangle has legs of length 22 feet and (30 - x) feet, so the hypotenuse of this triangle has length √(22² + (30 - x)²) = √(484 + (900 - 60x + x²)) = √(1384 - 60x + x²) feet.
Thus, the length of the wire is √(144 + x²) + √(1384 - 60x + x²) feet.