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Zorluk: OrtaIntegers, Absolute Value, and Number Lines

A hiker starts at a base camp at an elevation of 120120 feet. She hikes up to an overlook and then down to a valley at an elevation of −45-45 feet. The elevation of the overlook is higher than both the base camp and the valley. On a vertical number line representing elevations, the distance between the overlook's coordinate and the valley's coordinate is exactly twice the distance between the overlook's coordinate and the base camp's coordinate. What is the elevation, in feet, of the overlook?

  1. A
    65
  2. B
    70
  3. C
    95
  4. D
    195
  5. 285Cevap

Cevap

285 feet
The correct answer is 285285 feet. Let xx represent the elevation of the overlook. Because the overlook is higher than both the base camp at 120120 feet and the valley at −45-45 feet, we know that x>120x > 120. The distance between the overlook and the valley is x−(−45)=x+45x - (-45) = x + 45, and the distance between the overlook and the base camp is x−120x - 120. Setting up the relationship where the distance to the valley is twice the distance to the base camp gives the equation x+45=2(x−120)x + 45 = 2(x - 120). Solving this equation yields x+45=2x−240x + 45 = 2x - 240, which simplifies to x=285x = 285. This value is indeed higher than both elevations, and we can verify that the distance to the valley (285−(−45)=330285 - (-45) = 330) is twice the distance to the base camp (285−120=165285 - 120 = 165).

Adım Adım Çözüm

1
Define the variable and write the expressions for the distances on a number line.
Let xx represent the elevation of the overlook in feet. The distance between the overlook and the base camp (120120 feet) is ∣x−120∣|x - 120|. The distance between the overlook and the valley (−45-45 feet) is ∣x−(−45)∣=∣x+45∣|x - (-45)| = |x + 45|.
Distance between two points aa and bb on a number line is defined by the absolute value of their difference, ∣a−b∣|a - b|.
2
Set up the equation based on the given ratio.
∣x+45∣=2∣x−120∣|x + 45| = 2|x - 120|
The problem states that the distance to the valley is exactly twice the distance to the base camp.
3
Apply the condition that the overlook is higher than both locations to simplify the absolute value expressions.
Since x>120x > 120 and x>−45x > -45, both terms inside the absolute values are positive. Therefore, ∣x+45∣=x+45|x + 45| = x + 45 and ∣x−120∣=x−120|x - 120| = x - 120. The equation simplifies to: x+45=2(x−120)x + 45 = 2(x - 120).
For any expression y>0y > 0, ∣y∣=y|y| = y. Establishing the location of xx relative to 120120 and −45-45 allows us to remove the absolute value bars.
4
Solve the simplified linear equation for xx.
x+45=2x−240⇒x=285x + 45 = 2x - 240 \Rightarrow x = 285
Distribute the 22 on the right side and isolate xx by subtracting xx and adding 240240 to both sides.

Anahtar Kavram

Representing distances between points on a number line using absolute value equations.

Alternatif Yöntem

Instead of setting up equations, you can test the options starting from the constraint that the elevation must be greater than 120120 feet. This eliminates 6565, 7070, and 9595. Test the remaining options: for 195195, the distance to 120120 is 7575 and the distance to −45-45 is 240240 (not twice 7575). For 285285, the distance to 120120 is 165165 and the distance to −45-45 is 330330, which is exactly twice 165165.
Tahmini Süre:1m 30s
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Konular

Integers, Absolute Value, and Number Lines

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