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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A wheelchair ramp is constructed in two consecutive straight segments. The first segment rises at a 30∘30^\circ angle relative to the flat ground and has a length of 1212 feet. The second segment starts at the end of the first segment and rises at a 45∘45^\circ angle relative to the horizontal, with a length of 828\sqrt{2} feet. What is the total vertical rise, in feet, from the start of the first segment to the end of the second segment?

  1. A
    2222
  2. B
    6+826 + 8\sqrt{2}
  3. 1414Answer
  4. D
    8+638 + 6\sqrt{3}
  5. E
    16+6316 + 6\sqrt{3}

Answer

The total vertical rise is 1414 feet.
The total vertical rise is the sum of the vertical rises of the two individual segments. The first segment forms a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle where the hypotenuse is 1212 feet, so the vertical rise (the leg opposite the 30∘30^\circ angle) is half of the hypotenuse, which is 66 feet. The second segment forms a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle where the hypotenuse is 828\sqrt{2} feet, so the vertical rise (the leg opposite the 45∘45^\circ angle) is the hypotenuse divided by 2\sqrt{2}, which is 88 feet. Adding these two values gives a total vertical rise of 1414 feet.

Step-by-Step Solution

1
Calculate the vertical rise of the first segment.
The first segment has a length of 1212 feet and rises at a 30∘30^\circ angle. It forms a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle where the vertical rise is the side opposite the 30∘30^\circ angle. Since the leg opposite the 30∘30^\circ angle is half the hypotenuse, the vertical rise is 122=6\frac{12}{2} = 6 feet.
To find the vertical component of the first ramp segment.
2
Calculate the vertical rise of the second segment.
The second segment has a length of 828\sqrt{2} feet and rises at a 45∘45^\circ angle. It forms a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle where the leg length is the hypotenuse divided by 2\sqrt{2}. Thus, the vertical rise is 822=8\frac{8\sqrt{2}}{\sqrt{2}} = 8 feet.
To find the vertical component of the second ramp segment.
3
Sum the vertical rises of both segments.
The total vertical rise is 6 feet+8 feet=146\text{ feet} + 8\text{ feet} = 14 feet.
To find the combined vertical height gained over the entire ramp.

Key Concept

Using special right triangle ratios (30∘−60∘−90∘30^\circ-60^\circ-90^\circ and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ) to find missing side lengths.
Estimated Time:1m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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