Skier Lindsey Vonn completes a straight 300-meter downhill

BTGModeratorVI wrote: ↑

Sun Mar 15, 2020 12:37 pm

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

Answer: C
Source: Manhattan Prep

Time taken while downhill = Distance / Av. speed = 300/(x + 10) second;
Time taken while uphill = Distance / Av. speed = 300/(x – 8) second

Thus, 300/(x – 8) – 300/(x + 10) = 135

(x – 8)(x + 10) = 40

x = 10 meters/sec

Thus, av. speed during downhill = x + 10 = 10 + 10 = 20 meters/sec.

The correct answer: C

Hope this helps!

-Jay
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BTGModeratorVI wrote: ↑

Sun Mar 15, 2020 12:37 pm

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

Answer: C
Source: Manhattan Prep

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

BTGModeratorVI wrote: ↑

Sun Mar 15, 2020 12:37 pm

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

Answer: 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