Strange Calculus assignment - Related Rates? "Meredith goes on a hike..."

qdog83

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Hi there - My calculus prof gave us an assignment that has the whole class stumped. He specifically said that this assignment does not require calculus, and seems to be more of a riddle but i'm sure that it involves some calculus based concepts all things considered. Anyways, here's the assignment. Any and all help would be appreciated :) Thanks!

Meredith Mathie goes on a hike in the mountains, setting out from the Origin Lodge at 9 a.m. and taking a trail that winds over hill and dale through Maple Forest to Maximum Peak. On reaching the summit of Maximum Peak, Meredith immediately turns around and, following exactly the same trail in reverse, returns to Origin Lodge at 5 p.m. Meredith is a tad obsessive-compulsive and always walks at a speed of 4.5 km/h going uphill, 6 km/hon level ground, and 9 km/h going downhill. At no point between departing from and returning to Origin Lodge does Meredith take a break or pause.
1. What is the round-trip distance walked by Meredith? [5]
2. Determine, as accurately as you can, at what time Meredith reached the summit of Maximum Peak. [5]

p.s. no other information was given, this is all. Very confused where to start.
 
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Hi there - My calculus prof gave us an assignment that has the whole class stumped. He specifically said that this assignment does not require calculus, and seems to be more of a riddle but i'm sure that it involves some calculus based concepts all things considered. Anyways, here's the assignment. Any and all help would be appreciated :) Thanks!

Meredith Mathie goes on a hike in the mountains, setting out from the Origin Lodge at 9 a.m. and taking a trail that winds over hill and dale through Maple Forest to Maximum Peak. On reaching the summit of Maximum Peak, Meredith immediately turns around and, following exactly the same trail in reverse, returns to Origin Lodge at 5 p.m. Meredith is a tad obsessive-compulsive and always walks at a speed of 4.5 km/h going uphill, 6 km/hon level ground, and 9 km/h going downhill. At no point between departing from and returning to Origin Lodge does Meredith take a break or pause.
1. What is the round-trip distance walked by Meredith? [5]
2. Determine, as accurately as you can, at what time Meredith reached the summit of Maximum Peak. [5]

p.s. no other information was given, this is all. Very confused where to start.
Is the discussion here helpful? (Or here?) ;)
 
I would start with "rearranging" the trail to simplify things: divide it into 3 parts: 1. all downhill segments, 2. all level, 3. all uphill. Then use D = V*T to set up equations using the given numbers. I was able to calculate the round-trip distance. The other question is not very clear for now...
 
I get round-trip = 48 km and summit reached at 2.20 pm (5 1/3 hours).
S'that what you get Lev?
 
Easiest way is assume length of level path = 0.
It's length doesn't matter, since covered 50-50 by the 4.5 and 9 speeds.
 
Easiest way is assume length of level path = 0.
It's length doesn't matter, since covered 50-50 by the 4.5 and 9 speeds.

Why can we assume that? If the whole trail is level it would take 4 hours there and 4 back. Seems like a valid solution. If it's 100% uphill there and downhill back, then yes, it's 5.5 hours to the top. Based on the problem statement can't there be any number of uphill, downhill and level segments?
 
It is obvious from this that path cannot be fully level (notice words "over" and "summit":

"Meredith Mathie goes on a hike in the mountains, setting out from the Origin Lodge at 9 a.m. and taking a trail that winds "over" hill and dale through Maple Forest to Maximum Peak. On reaching the "summit" of Maximum Peak"......

Then the speeds are given; nowhere does it state that ALL were used:

"Meredith is a tad obsessive-compulsive and always walks at a speed of 4.5 km/h going uphill, 6 km/h on level ground, and 9 km/h going downhill."

Thus assuming no "level" portions seems ok....

No?
 
Taking x = distance that is level, y = distance that is uphill on the way, and z = distance that is downhill on the way, I get x + y + z = 24, so the round-trip distance is 48 km as others have said.

But the time to the peak is x/6 + y/4.5 + z/9 = (3x+4y+2z)/18, which is not independent of the breakdown. The least it can be is 24/9 = 2 2/3 hr (all downhill), and the greatest it can be is 24/4.5 = 5 1/3 hr (all uphill), ignoring how the distances can be broken down and actually call it a peak. Presumably (unless there can be variations in slope) y>z and x<24, so the time must be more than 4 hr (all level or equal up and down), if I'm thinking correctly. But it can be anything in 4 < t < 5 1/3 hr.

Note the wording of the question: "Determine, as accurately as you can". The answer will, I think, be an interval -- we can't be more accurate than that.

I suspect the idea that it can be assumed that it is never level may come from assuming there is a single answer. Am I right?
 
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Yes, I agree with 4 < t < 5 1/3 hr. I mentioned the all level case to point out that there are multiple solutions.
 
Thanks for all the help everybody. My problem was I was searching for a single answer which has now been made obvious doesn't exist. Searching for an interval seems like a much more effective solution. Thanks again!
 
How did you find this number?

I get round-trip = 48 km and summit reached at 2.20 pm (5 1/3 hours).
S'that what you get Lev?


I think I am in the same calc class and I am currently struggling with the same assignment. How did you find these answers
 
help

Taking x = distance that is level, y = distance that is uphill on the way, and z = distance that is downhill on the way, I get x + y + z = 24, so the round-trip distance is 48 km as others have said.

But the time to the peak is x/6 + y/4.5 + z/9 = (3x+4y+2z)/18, which is not independent of the breakdown. The least it can be is 24/9 = 2 2/3 hr (all downhill), and the greatest it can be is 24/4.5 = 5 1/3 hr (all uphill), ignoring how the distances can be broken down and actually call it a peak. Presumably (unless there can be variations in slope) y>z and x<24, so the time must be more than 4 hr (all level or equal up and down), if I'm thinking correctly. But it can be anything in 4 < t < 5 1/3 hr.

Note the wording of the question: "Determine, as accurately as you can". The answer will, I think, be an interval -- we can't be more accurate than that.

I suspect the idea that it can be assumed that it is never level may come from assuming there is a single answer. Am I right?

Can you explain the math you did to get x+y+z=24
 
Finding the distance?(Not sure if this correct)

t1= time from Op to Mp
t2=time from (Op to Mp) twice
d1= Uphill
d2= Ground level
d3= Downhill
t1+t2=8

(d1/4.5)+(d2/6)+(d3/9)=t1
(d3/4.5)+(d2/6)+(d1/9)=t2

((d1+d3)/4.5)+(2d2/6)+((d1+d3)/9)=8

(d1+d3)*((2/9)+(1/9))+(d2(1/3))=8

(d1+d3)*(1/3)+(d2(1/3))=8

From here I recognize that nothing can be done, except making assumptions for (d1+d3), which does seem quite ideal, since it asking to find the distance.

Can anybody help me out!!

Thanks in Advance!!
 
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