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Skier Lindsey Vonn completes a straight 300-meter downhill

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Skier Lindsey Vonn completes a straight 300-meter downhill

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Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

OA C

Source: Manhattan Prep

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BTGmoderatorDC wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30

OA C

Source: Manhattan Prep
We can let the time going down = 300/(x +10) and the time going up as 300/(x - 8), thus:

300/(x +10) + 135 = 300/(x - 8)

Multiplying by (x + 10)(x - 8), we have:

300(x - 8) + 135(x + 10)(x - 8) = 300(x + 10)

20(x - 8) + 9(x + 10)(x - 8) = 20(x + 10)

20x - 160 + 9(x^2 + 2x - 80) = 20x + 200

20x - 160 + 9x^2 + 18x - 720 = 20x + 200

9x^2 + 18x - 1,080 = 0

x^2 + 2x -120 = 0

(x + 12)(x - 10) = 0

x = -12 or x = 10

Since x can’t be negative, x must be 10. Therefore, the speed for the downhill run was 10 + 10 = 20 meters per second.

Answer: C

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BTGmoderatorDC wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30
Downhill speed - uphill speed = (x+10) - (x-8) = 18.
Implication:
The downhill speed is 18 mps greater than the uphill speed.

We can PLUG IN THE ANSWERS, which represent the downhill speed.
Since the downhill speed is 18 mps greater than the uphill speed, the correct answer must be GREATER THAN 18.
Eliminate A and B.

The remaining answer choices imply the following:
C: downhill speed = 20, uphill speed = 20-18 = 2
D: downhill speed = 25, uphill speed = 25-18 = 7
E: downhill speed = 30, uphill speed = 30-18 = 12
Since the value in red does not divide evenly into the 300-meter distance, D is almost certain to be incorrect.
When the correct answer is plugged in, uphill time - downhill time = 135 seconds.

C: downhill speed = 20, uphill speed = 20-18 = 2
Time to travel 300 meters uphill at a speed of 2 mps = 300/2 = 150 seconds.
Time to travel 300 meters downhill at a speed of 20 mps = 300/20 = 15 seconds.
Uphill time - downhill time = 150-15 = 135 seconds.
Success!

The correct answer is C.

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BTGmoderatorDC wrote:
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x - 8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?

(A) 10
(B) 15
(C) 20
(D) 25
(E) 30
The ride up the mountain took 135 seconds longer than her run down the mountain
Start with a word equation: (time going UP mountain) = (time going DOWN mountain) + 135
time = distance/speed
We can now write: 300/(x - 8) = 300/(x + 10) + 135
Multiply both sides by (x - 8) to get: 300 = 300(x - 8)/(x + 10) + 135(x - 8)
Multiply both sides by (x + 10) to get: 300(x + 10) = 300(x - 8) + 135(x - 8)(x + 10)
Divide both sides by 5 to get: 60(x + 10) = 60(x - 8) + 27(x - 8)(x + 10)
Divide both sides by 3 to get: 20(x + 10) = 20(x - 8) + 9(x - 8)(x + 10)
Expand both sides to get: 20x + 200 = 20x - 160 + 9x² + 18x - 720
Rearrange and simplify to get: 9x² + 18x - 1080 = 0
Divide both sides by 9 to get: x² + 2x - 120 = 0
Factor to get: (x + 12)(x - 10) = 0
So, EITHER x = -12 OR x = 10
Since x can't be the speed, we know that x = 10

What was her average speed, in meters per second, during her downhill run?
Her downhill speed = x + 10
Since x = 10, her downhill speed = 10 + 10 = 20

Answer: C

Cheers,
Brent

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