Once upon a cache Mystery Cache
The_Incredibles_: And we're done.
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Difficulty:
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Terrain:
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Size: 
(small)
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This puzzle cache requires the use of calculus. There are many online tools and information to help you. Graphing the functions will help. The difficulty rating is set high due to the subject matter, but the puzzle shouldn't be too hard if you understand the basics and use the online tools available. Please note: log(x) should be interpreted as the natural logarithm, ln(x) not log10(x).
The cache is located at N 48 AB.CDE W 123 FG.HIJ.
I got the idea for this puzzle cache a while ago, but I didn't start working on it until I had a conversation with cameracowgirl the other day. She's taking a math class right now and is planning a puzzle cache to celebrate all that she's learned. She'll also be taking a calculus course soon so I thought it would be fun to make a calculus puzzle cache to give her something extra to do. The jokes have nothing to do with the puzzle; they are there purely for your amusement. Have fun!
Question 1
It was a cold and rainy day in Victoria and THE GEOKNIGHTS were driving south down the Pat Bay. Their phone rang - a new cache was published - the race was on! Well known for their efficiency in the FTF race, they promptly engaged geo-warp speed. Instantly, they were traveling 8.0 geo-warps/hour with only 12 km to go. After much zigging and zagging, they were in the right area. Fortunately the cache was hidden along a straight stretch of road so as their cachemobile sped by GZ, they were able to jump out, roll into the bushes and have the cache in hand within seconds. Another FTF!
Their distance to the cache is represented by the following equation:
where T is the time in minutes passsed since notification, 0 < T < 5.8
Please determine the speed their cachemobile was traveling when they reached GZ, to 4 decimal places. Please note this will come up as a negative number due to the zig-zagging they had to do to get to GZ.
Speed (geo-warps) upon arrival at GZ = - _ _ . _ (H-2) _ A
Question 2
The GO! Team was out and about, enjoying the sunshine and picking up a few caches here and there. One of the caches they stopped to find was at the base of a tree next to a busy walkway along the sea. Despite the abundance of ivy, their geosenses were tingling and the cache was soon in hand. They paused to sign the logbook, admire the swag and take in the view. They found a bouncy ball and started to play catch with it. No sooner had they started, when a jogging muggle collapsed on a nearby bench. Always happy to help a muggle in distress, the GO! Team dropped the bouncy ball and rushed to help. In their haste, they failed to notice the ball roll, drop into the water, and later, resurface.
The depth of the bouncy ball is given by the following equation:
where T is the time in seconds passed after the bouncy ball enters the water, 0 < T < 0.6
Please calculate the depth of the ball when it's downward speed equals 0.7 m /s, to 4 decimal places.
Depth at speed of 0.7 m/s = _ . (D-4) _ (C+1) _ meters
Question 3
Cameracowgirl finished all her homework early one night so she decided to take a look at how her BBQ travel bug was progressing. She was excited to see that a hard-working lackey had added some new functions. With a joyful click of her mouse button, she was able to see a bar graph of the total distance travelled for each month since her bug had been released. To her further delight, one more click of her mouse revealed a moving average and a 'best-fit' function for the data. Eager to meet the mastermind behind the new functions, she dropped everything she was doing, drove directly to the airport, took the next plane to Seattle and hasn't been seen since.
The 'best fit' function is represented by:
where 0 < T < 18 months
Please find the maximum of the moving average of the # of km travelled/month, according to the best fit function given, to 4 decimal places.
Maximum = (J-1) _ . _ _ (E+1) F
Question 4
It was as sunny Saturday afternoon and a new cache was published. First to arrive at the trailhead were local FTF hounds, Mr islandfinders and the A-Team. If it were any other cache, they would have had a good chat and a leisurely stroll on the way to GZ. This cache was different, however. It had $200 in it and their hydro bills were due!! After a grunt and a nod, both cachers started to RUN down the trail as fast as they could. Dodging tree branches and hopping over roots and puddles, they dashed through the forest. Neck and neck, they ran along the trail, each hoping for a FTF. Soon, they rounded a bend in the trail and came to a wide, straight stretch. They spotted the cache 50 meters away as the cache container was a large pinecone hanging high in a tree. When they got close enough, they both jumped for it, hoping to be the first to touch the cache. As luck would have it, their hands made contact with the cache at precisely the same time.
The trajectory of Mr islandfinders' hand is given by the following equation:
islandfinders:
where h is the height of his hand above ground and x is the horizontal distance his hand has travelled since the beginning of his leap (both in meters, 0 < x < 8)
Similarly, for the A-Team:
Also, the height of Mr islandfinders' hand above ground is given by the following equation:
where h is the distance in meters of his hand above ground and T is the number of seconds elapsed since the start of his leap.
Please find the vertical component of the velocity of Mr islandfinders' hand at the time his hand touches the cache, to 4 decimal places:
Vertical component of velocity (m/s) = _ . B _ (G-2) I
Additional Hints
(Decrypt)
9-1 6-3 5-3 3-3
7-3 2-1 6-1
2-1 5-3 7-1 4-2 2-1
(guvax grkgvat: 2-1 rdhnyf N)
Ab arrq gb qb nal bs gur zngu ol unaq. Rknzcyrf sbe hfvat gur nobir gbby:
1) Gb tencu na rdhngvba, glcr:
"cybg l = k^2"
2) Gb svaq n qrevingvir, glcr:
"svaq qrevingvir bs l=k^2"
3) Gb svaq gur k-vagreprcgf, glcr:
"svaq k-vagreprcgf bs l = fva(k)"
4) Gb fbyir sbe k, glcr:
"svaq k vs 0=k^2+9*k+19"
5) Gb rinyhngr na rkcerffvba, glcr:
"rinyhngr l = k^2 sbe k = 5"
Svany: Haqre zbffl ybt, genvy vf n ovg biretebja, npprff sebz Abegu-Rnfg bs gur pnpur pbbeqvangrf.
Treasures
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