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Difficulty: MediumPythagorean Theorem Word Problems

A telecommunications crew is installing a straight fiber-optic cable from the base of a network tower to a control panel mounted on a nearby building. The horizontal distance from the tower base to the building is 1212 yards. The control panel is mounted 55 yards above the ground. How many feet long must the cable be to reach exactly from the tower base to the control panel?

  1. A
    13
  2. B
    17
  3. 39Answer
  4. D
    51

Answer

39
The scenario creates a right triangle with legs measuring 55 yards and 1212 yards. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we find the hypotenuse is 52+122=25+144=169=13\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 yards. Because the question asks for the answer in feet, we multiply 1313 yards by 33 feet per yard to get exactly 3939 feet.

Step-by-Step Solution

1
Identify the shape formed by the scenario.
The ground, the building wall, and the cable form a right triangle.
This allows the use of the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the length of the diagonal cable.
2
Calculate the length of the cable in yards.
52+122=25+144=1695^2 + 12^2 = 25 + 144 = 169. The square root of 169169 is 1313 yards.
Applying the Pythagorean theorem using the given leg lengths of 55 yards and 1212 yards yields the hypotenuse.
3
Convert the final length from yards to feet.
13 yards×3 feet/yard=39 feet13 \text{ yards} \times 3 \text{ feet/yard} = 39 \text{ feet}.
The question specifically asks for the length of the cable in feet.

Key Concept

Pythagorean Theorem and Unit Conversions
Estimated Time:1m 0s
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Topics

Pythagorean Theorem Word Problems

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