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Difficulty: HardTriangles and Pythagorean Theorem

A vertical flagpole that is originally 3232 feet tall breaks at a certain height due to strong winds. The top part of the pole folds over without completely detaching, and the tip touches the level ground exactly 1616 feet away from the base of the pole. At what height from the ground did the flagpole break?

  1. A
    8 feet
  2. 12 feetAnswer
  3. C
    16 feet
  4. D
    20 feet

Answer

12 feet
The scenario creates a right triangle where the vertical leg is the standing part of the pole, the horizontal leg is the ground, and the hypotenuse is the broken top part. Letting the break height be xx, the hypotenuse becomes (32−x)(32 - x). Using the Pythagorean theorem: x2+162=(32−x)2x^2 + 16^2 = (32 - x)^2. Expanding the right side gives x2+256=1024−64x+x2x^2 + 256 = 1024 - 64x + x^2. Subtracting x2x^2 from both sides simplifies the equation to 256=1024−64x256 = 1024 - 64x. Adding 64x64x and subtracting 256256 gives 64x=76864x = 768. Dividing by 6464 yields x=12x = 12. Therefore, the pole broke 1212 feet above the ground.

Step-by-Step Solution

1
Define the variables for the sides of the right triangle formed by the broken flagpole.
Let the height of the break (standing part) be xx. The length of the fallen part (hypotenuse) is the remainder of the pole, which is (32−x)(32 - x). The base distance along the ground is 1616 feet.
Establishing these algebraic relationships is necessary to apply the Pythagorean theorem to the scenario.
2
Set up the Pythagorean theorem equation (a2+b2=c2a^2 + b^2 = c^2) using the defined variables.
x2+162=(32−x)2x^2 + 16^2 = (32 - x)^2
The standing pole and the ground form a right angle, making the standing pole and ground distance the legs, and the fallen part the hypotenuse.
3
Expand both sides of the algebraic equation.
x2+256=1024−64x+x2x^2 + 256 = 1024 - 64x + x^2
Expanding the squared binomial (32−x)2(32 - x)^2 correctly is required to isolate the variable xx.
4
Solve for xx by combining like terms.
Subtract x2x^2 from both sides to get 256=1024−64x256 = 1024 - 64x. Rearrange to 64x=76864x = 768, and divide by 6464 to find x=12x = 12.
Isolating xx provides the exact height from the ground where the flagpole snapped.

Key Concept

Applying the Pythagorean Theorem with algebraic expressions
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Topics

Triangles and Pythagorean Theorem

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