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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A vertical flagpole stands perpendicular to level ground. Two support cables are attached from the top of the flagpole to ground anchors located on opposite sides of the pole. The first cable makes a 60∘60^\circ angle with the ground and its ground anchor is 10310\sqrt{3} feet from the base of the pole. The second cable makes a 45∘45^\circ angle with the ground. What is the sum of the lengths, in feet, of the two support cables?

  1. 203+30220\sqrt{3} + 30\sqrt{2}Answer
  2. B
    103+30210\sqrt{3} + 30\sqrt{2}
  3. C
    303+30230\sqrt{3} + 30\sqrt{2}
  4. D
    203+3020\sqrt{3} + 30
  5. E
    50550\sqrt{5}

Answer

203+30220\sqrt{3} + 30\sqrt{2} feet
The first support cable forms a 30°-60°-90° right triangle with the flagpole and ground. Given the adjacent leg to the 60° angle is 10310\sqrt{3} feet, the opposite leg (the flagpole's height) is 103×3=3010\sqrt{3} \times \sqrt{3} = 30 feet, and the hypotenuse (first cable) is 2×103=2032 \times 10\sqrt{3} = 20\sqrt{3} feet. The second cable forms a 45°-45°-90° right triangle sharing the 30-foot height as one leg. Thus, the hypotenuse (second cable) is 30230\sqrt{2} feet. Adding both cable lengths gives 203+30220\sqrt{3} + 30\sqrt{2} feet.

Step-by-Step Solution

1
Analyze the first right triangle formed by the pole and the first cable
The triangle is a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle with the ground angle equal to 60∘60^\circ. The side adjacent to 60∘60^\circ (shorter leg) is 10310\sqrt{3} ft.
The cable makes a 60∘60^\circ angle with the horizontal ground, making the angle at the top of the pole 30∘30^\circ.
2
Calculate the height of the flagpole and the length of the first cable
Flagpole height =103⋅3=30= 10\sqrt{3} \cdot \sqrt{3} = 30 ft. First cable length (hypotenuse) =2⋅103=203= 2 \cdot 10\sqrt{3} = 20\sqrt{3} ft.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle with shorter leg xx, the longer leg is x3x\sqrt{3} and the hypotenuse is 2x2x.
3
Calculate the length of the second cable using the second right triangle
Second cable length (hypotenuse) =302= 30\sqrt{2} ft.
The second triangle is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle with leg equal to the pole height (3030 ft). The hypotenuse is leg⋅2\text{leg} \cdot \sqrt{2}.
4
Add the lengths of the two support cables
Total length =203+302= 20\sqrt{3} + 30\sqrt{2} ft.
Summing the two hypotenuse lengths gives the total combined cable length.

Key Concept

Properties of 30∘−60∘−90∘30^\circ-60^\circ-90^\circ and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ special right triangles
Estimated Time:1m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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