Skier Lindsey Vonn completes a straight 300-meter downhill

This is way more simpler than the solution offered by MGMAT! Thanks

can anyone explain step by step how from this 300/(x+10) + 135 = 300/(x-8) we got this x^2+2x-120 =0

dave13, this is a great example of patience ( Delaying the simplification by opening brackets)

\(\frac{300}{t} = (x + 10)\) ... this is the distance time speed relation from her run downhill.

\(\frac{300}{(t+135)} = (x - 8)\) ... this second relation comes from her chairlift uphill.

We need to solve for x and hence from the above two equations we need to eliminate t.

Let's substitute the value of \(t = \frac{300}{(x+10)}\) in the second equation. Yes, it looks scary but things will get better now

\(\frac{300*(x + 10)}{(300+(x+10)*135)} = x - 8\)
\(300*(x+10) = 300*(x-8) + 135*(x+10)*(x-8)\) Let's not start opening brackets yet as things will reduce.
\(60*(x+10) = 60*(x-8) + 27*(x+10)*(x-8)\) 5 is common in all numbers hence it has been cancelled out
\(20*(x+10) = 20*(x-8) + 9*(x+10)*(x-8)\) Now 3 is common'

Now we are ready to open brackets and simplify

\(20x + 200 = 20x - 160 + 9(x+10)(x-8)\)
\(360 = 9*(x+10)(x-8)\) 9 can still cancel from both sides
\(40 = x^2 + 2x - 80\) Finally opening the brackets
\(x^2 + 2x - 120 = 0\)

Easy quadratic to solve now

\(x^2 + 12x - 10x - 120 = 0\)
\(x(x+12)-10(x+12) = 0\)
\((x-10)*(x+12) = 0\)

Hence x = 10 discarding the negative root

Hence the speed downhill was ( x + 10) or 20 /s

Key takeaway here is : smart calculation saves a lot of time

Best,
Gladi

I also got stuck with the algebra....its not difficult but it is LONG.
my takeaway: if you are sure of the basic setup of the equation AND it looks that it will get ugly, then it is faster and CLEANER to plug the answer choices than to solve for roots.

this is something i have to remind myself each time i start a question....to get into the habit.

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

RELATED VIDEO

The conventional explanation involving quadratic equation is quite long. Fortunately, in this particular problem, there is a much quicker way using the answer choices.
Lindsay's uphill speed is 18 m/s less than her downhill speed so the latter must be more than 18 m/s otherwise her uphill speed will be negative which is not possible. So options (A) and (B) are out. Let's consider option (C):
Her downhill speed is 20 m/s; therefore, her uphill speed is 2 m/s and the time she takes for her downhill run is 300/20=15 seconds. The time taken for her uphill trip is thus (15+135)=150 seconds. So we see that option (C) checks out because she can cover travelling 150 seconds at 2 m/s she covers the 300 meters of her ski run.

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

Downhill run:
Speed = (x+10) m/s
Distance = 300 m
Time taken = 300/(x+10) seconds

Uphill run:
Distance = 300 m
Speed = (x-8) m/s
Time taken = 300/(x-8) seconds

Time difference between uphill and downhill run = 135 seconds = 300/(x-8) - 300/(x+10) = 300 * 18 / (x-8)(x+10)

(x-8)(x+10) = 300*18/135 = 60*2/3 = 40 = 2*20
x = 10

Average speed during downhill run = x + 10 = 20 m/s

IMO C

We can skip the algebra altogether and Plug In The Answers (PITA).

x has to be greater than 8 or she will be going backward on the chairlift. x+10 is therefore greater than 18. A and B are out.

Let's plug in D.
She skis down at 25mps. 300 meters will take 12 seconds.
She skis down at x+10, so x=15.
She rides up at x-8, so 7mps. 300 meters will take somewhere around 40-45 seconds.
That's a difference of way too little time. D is out.

It might not be obvious whether we need something smaller or something larger, but we only have C and E remaining, so we can try either. Let's do C.
She skis down at 20mps. 300 meters will take 15 seconds.
She skis down at x+10, so x=10.
She rides up at x-8, so 2mps. 300 meters will take 150 seconds.
That's a difference of 135 seconds. Yay!

Answer choice C.


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