Examkin

Question

Difficulty: MediumPythagorean Theorem and Special Right Triangles

A mountain zipline course consists of two connected ascending sections. The first section starts at base station AA and rises to intermediate platform BB at an angle of 30∘30^\circ relative to the horizontal ground, covering a horizontal distance of 40340\sqrt{3} meters. The second section rises from platform BB to peak platform CC at an angle of 45∘45^\circ relative to the horizontal, covering a horizontal distance of 3030 meters. What is the total vertical height, in meters, of peak platform CC above base station AA?

  1. A
    4040
  2. 7070Answer
  3. C
    150150
  4. D
    203+3020\sqrt{3} + 30
  5. E
    403+3040\sqrt{3} + 30

Answer

The total vertical height of peak platform C above base station A is 70 meters.
To find the total vertical height of peak platform C above base station A, determine the vertical rise of each section individually using special right triangle rules and sum them together. For the first section, the 30∘30^\circ incline forms a 30∘30^\circ-60∘60^\circ-90∘90^\circ right triangle where the horizontal distance of 40340\sqrt{3} meters is adjacent to the 30∘30^\circ angle. The vertical rise is opposite the 30∘30^\circ angle, so dividing 40340\sqrt{3} by 3\sqrt{3} gives a vertical rise of 4040 meters. For the second section, the 45∘45^\circ incline forms a 45∘45^\circ-45∘45^\circ-90∘90^\circ right triangle where the vertical leg equals the horizontal leg, giving a vertical rise of 3030 meters. Adding these two vertical heights yields 40+30=7040 + 30 = 70 meters.

Step-by-Step Solution

1
Calculate the vertical rise of the first section using 30∘30^\circ-60∘60^\circ-90∘90^\circ special right triangle properties.
The vertical height of the first section is 4040 meters.
In a 30∘30^\circ-60∘60^\circ-90∘90^\circ triangle, the ratio of the side opposite the 30∘30^\circ angle (vertical rise) to the side opposite the 60∘60^\circ angle (horizontal distance) is 1:31 : \sqrt{3}. Dividing the horizontal distance 40340\sqrt{3} by 3\sqrt{3} yields 4040 meters.
2
Calculate the vertical rise of the second section using 45∘45^\circ-45∘45^\circ-90∘90^\circ special right triangle properties.
The vertical height of the second section is 3030 meters.
In an isosceles right triangle (45∘45^\circ-45∘45^\circ-90∘90^\circ), the two legs are equal in length. Since the horizontal distance is 3030 meters, the vertical rise is also 3030 meters.
3
Sum the vertical rises from both sections to find the total vertical height.
Total height = 40+30=7040 + 30 = 70 meters.
The total vertical elevation of point C above point A is the sum of the vertical changes along each segment of the path.

Key Concept

Special Right Triangle Side Ratios (30∘30^\circ-60∘60^\circ-90∘90^\circ and 45∘45^\circ-45∘45^\circ-90∘90^\circ)
Estimated Time:1m 30s
Rate this question

Topics

Pythagorean Theorem and Special Right Triangles

More questions from this topic