NeoPets Riddles (Lenny Conundrums) and Answers Here

i found my error. i was using 1,2 m for the side measure. thank you anyway

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Can't we pretend as if there's enough ice? Because it doesn't say we have to stick with the amount of ice that is given, right?

Because then we can count further... There's a volume of 0,48771m^3 that we need to count with. I assume it is the same and it melts the same as the sphere.
So 4/3*pi*r^3 = 0,48771. r = 0,4883.
And that makes 0,4883/0,0015 = 325,54655 minutes that we have to add up with the 666,6667 minutes, which makes 992 minutes rounded.

Right?

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Part 1

The volume of a cylinder is the product of the area of the base and the height. The base is a hexagon, which are 6 equilateral triangles with a side of 1.5 m. The area (A) of that base is 6*2/3bh m^3.

b=1.5 m, h=3^(1/2)/2*1,5 m so
A=6*(1/2)*1.5* 3^(1/2)/2*1,5=3*3^(1/2)/2*(1.5)^2 m^2

The height of the cylinder is 0.8m, so the volume is
Vc=0.8*3*3^(1/2)/2*(1.5)^2 m^3=4.6765 m^3

The ice sphere is 2m in diameter (radius (r) is 1m), so the volume of the sphere is
Vs=4/3*pi*r^3=4.1888 m^3

so the bath will be 89.6% full, but the volume of the standard female, based on a weight of 122 lbs with the density of 60 pounds/ft^3) is 0.0566 m^3

With the princess the bath will be 90.8% full filling the tub to a depth of 0.726 m.

Part 2

Ice melting time

The Volume of the sphere is

Vs=4/3*pi*r^3
So
dVs/dr=4*pi*r^2
And the melting rate is 1.5 mm/min, which is equal to 1.5/1000 m/min. The melting reduces the diameter so this melting rate is negative with the boundary condition that at zero time (t=0) the sphere radius is 1m.
Or
dVs/dt=dVs/dr*dr/dt=-1.5/1000*4*pi*r^2
@ t=0, r=1
Integrating
Vs=-1.5/1000*4/3*pi*t+4/3*pi=4/3*pi(1-1.5/1000*t)
When the ball is completely melted Vs=0
So
0=4/3*pi(1-1.5/1000*t) or
1-1.5/1000*t=0
And t=1000/1.5 min=667 min

So the ice ball will take 11 hours and 7 minutes to melt (and will it be cold)

Now to heat the princess' bath (to a reasonable 90 degrees) will take 830,000 btu. Using a reasonably sized hot water heater used to start the ice ball melting (about 1,600 Btu/m) will take an additional 500 minutes (6:20).

BTW the thermal expansion of water will fill the tub closer to full.

So in about 17:21 the princess can have a comfortable bath.

Rap

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Hi I`m new to the board. I've done the calculations and i'll post them for you to compare. I'm not posting the obvious formulas, we got enough of those!

Area of the hexagon = 5.846 m^2
Vh = 5.846 * 0.8 = 4.6768 m^3

-->4.6768 m^3 is the area to be filled

INITIAL volume of the sphere:

Vi = 4.1888 m^3

END volume of the sphere after melting:

Ve = 4/3*pi*( 1 - 0.0015*t) ^3

where t is the time elapsed in minute.

THE MAIN FORMULA IS:
Vh = Vi - Ve

rule of 3, just replace Smile

4.6768 = 4.1888 - ( 4/3*pi*( 1 - 0.0015*t) ^3 )

a solve gave me 341.066 minutes

I think my formula is good but my calculator just can't take it.

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Wait, i was wrong, i assumed there would be ice left. But i just realized there isnt enough ice... I'm such a goof.

Ok sorry, i took the time to read through the WHOLE board.

I think the volume of the current female is a little far fetched... especially since we aren't talking about a human here. The princess is a aisha right? If you lookup a aisha, it will give you it's weight and height, but you can't get it's volume from that.

Also, the desert temperature is 56 celcius. We can suppose she doesn't heat the bath but that the water will go from 0 to 56 celcius right?

Rap, can you explain why you use intergral? I'm just starting to learn those and i don't see their purpose here.

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x wrote:

Nique, you're assuming that the tub is "completely full" when the ice sphere melts. I think a more likely assumption is what most others have made, that it's "completely full" when it's filled to the brim with water.

I didn't say that it's completely full when the sphere is done melting, I said that with the information given they most likely want how long the ice takes to melt, since it doesn't mention any extraneous ice. I also said that most of us don't fill a bath completely to the brim because some of the water would be displaced and wasted, adding once more to my hypothesis that they only want the amount of time it would take for the ice to melt.

As for the temperature of the ice, we don't know what temperature the ice was to begin with (it could have been -40 degrees for all we know) and calculating the time it would take to heat the water to desert temperature would take too long and probably be too complicated for a majority of Neopets players, even though I realize that not all players play the Lenny Conundrum. TNT usually gives us all the information we're supposed to use in these, don't they? They don't normally require out-of-the-box thinking such as "if it takes x time to fill y metres of water in the bath, then y/[depth of bath] / x is the answer." (A likely inacurrate equation, my brain exited math mode when I passed it.) If you can show me a couple of conundrums where they have required you to extrapolate, I'll be glad to retract this, but as is, I honestly think they expect us to realize that the bath's volume is more than the sphere's and only calculate the time it takes for the sphere to melt. In fact, I'm almost sure that the volumes are only there to confuse people. Does TNT ever make them that tough to solve?

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Quote:

As for the temperature of the ice, we don't know what temperature the ice was to begin with (it could have been -40 degrees for all we know)
(...)
TNT usually gives us all the information we're supposed to use in these, don't they?

The ice melts at 0 degrees. Since the timer starts from when the ice STARTS to melt, don't worry about the temperature of the ice.

And i think you are right, besides some constants from laws of physics or chemistry, i don't think they can ask us to get any alternate sources. I got the 56 degree temperature from the weather report.
http://neopets.com/weather.phtml?world=2

but i don't know how far they want us to go in our thinking... I doubt the answer is "Never"

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The ice melts at 0, yes (a fraction above 0, but I'm not that picky) but the temperature of the ice would affect the temperature of the air around the ice and, by proximity, the amount of energy left to heat the water, if you wanted to get technical. ^_~ Just making the point that we should only pay attention to the information that's given in the question rather than trying to guess what's on a point beyond the scope of the given data.

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Nique wrote:

Does TNT ever make them that tough to solve?

This is probably the hardest one, besides a joke one they had a while back that was all gibberish [and absolutely unsolvable]:

Quote:

Round: 99

The writer of the Lenny Conundrums decided, as a New Year's resolution, to make the Lenny Conundrums a little more challenging, because it seemed like everyone was complaining abut how easy the puzzles were.

His second New Year's resolution was to learn a new hobby. The hobby he chose was structural engineering. So he enrolled in the Brightvale Institute of Technology and Engineering. While studying there, he observed a new building being constructed.

There was a steel beam 50 feet long. The entire Skeith construction crew (total weight: 6000 pounds) was standing at the center of the beam. Assume the modulus of elasticity is 29,000,000 pounds per square inch, the moment of inertia is 850 inches^4, the beam has simple pin connections at either end, and all loads other than the weight of the crew are disregarded.

How much will the beam deflect? Please round to the nearest tenth of an inch.

Answer: -1.1 inches

The formula for beam deflection with simple supports and a point load at the center is PL3/ 48EI, where P is the load, L is the total length of the beam, E is the modulus of elasticity, and I is the moment of inertia.

So in this example, P = -6000 pounds (it's negative because it's a downward force), L = 600 inches (need to convert from feet to inches keep the units consistent!), E = 29000000 pounds per square inch, and I = 850 inches4.

So, plug them all into the equation, and you get about -1.1 inches.

Hopefully, we won't still be getting any complaints about how easy the Lenny Conundrum is!

There were even folks who complained because they forgot the negative, after all that Confused

idelhigh wrote:

I think the volume of the current female is a little far fetched... especially since we aren't talking about a human here. The princess is a aisha right? If you lookup a aisha, it will give you it's weight and height, but you can't get it's volume from that.

I think it would be unfair to include the Princess in the formula, because there's no way to determine THAT Aisha's volume, not accurately. Even looking up random Aishas on the site isn't accurate, because Neopets' volumes stay the same even if their species changes... Shocked

Yeah, I've had some tiny Grarrls [tyrannosaurs] and HUMONGOUS Pteris [sparrows]! Laughing

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Correction of sphere melting (thanks to idlehigh)

Integration, in this case is probably unnecessary. The ice melting rate is given as a penetration rate (1.5mm/min) results in a variable volumetric rate of ice melting.

Look at it this way. The surface area of a sphere is given by
Sa=4*pi*r^2
The melting rate given as 1.5mm/min is effectively the rate the radius decreases--initially.

So the initial rate of ice melting as given by
Q=Sa*dr/dt=4*pi*r^2*dr/dt is when the sphere has a 1 m radius.

This is
Q=0.0188 m^3/min
Cause when the sphere is half that diameter, the rate of ice melting is
Q= 0.0047 m^3/min

IOW the melting rate in terms of volume is variable. This won't happen if the temperature is constant. The amount of ice melted per minute is more likely to be constant as the ambient temperature is constant.

Now, is you assume that the initial melting rate is constant (0.0188 m^3/min) the sphere with a volume of 4.6768 m^3 will melt in about 249 minutes.

Rap

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The question specifically says that it melts constantly at 1.5 mm per minute, Rap. "If the desert heat uniformly melts the surface of the ice at a rate of 1.5mm of depth per minute..." Again, I point out that the only Conundrum that's involved complicated formulae was number 99, and that was mostly a joke. Neopets tends to follow the KISS rule: Keep It Simple, S[illy].

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okay.

if r is the radius of the ice in terms of the time melting,
r = 1 - .0015t

when r = 0, the ice has completely melted.
0 = 1 - .0015t
1 = .0015t
t = 1/.0015 = 667

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Not quite simplified the way I would have done, but yeah, that's what I've been saying. Once you realize that the ice sphere has to melt completely to come close to filling the tub to the brim, and when you factor in that Neopets did this deliberately and don't mention that you can add more ice since they don't say that any more is available to the princess, 1/0.015 is the answer, although that was stated pages back. ^^

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Still doubts about the fact the tub is not full by the time the ice melted... Confused

But... most lenny conundrums aren't too mathematic, for those who don't like calculating that much, right? So if I look at it that way, the answer has to be 667, which is quite an easy sum...

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Ok so i agree that it will take 667 minutes for the ice ball to completly melt.

But the tub wont be filled... and the question is "how many minutes will it take for the bath to be completly full". So that's the real problem. Is it a trick question or are we missing something? Would they give us the tub information for nothing?

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They gave us the tub information to confuse us, I think. I mean, you'd read it, calculate the volumes, realize that the sphere < the tub, and think that they miswrote the question or something. Since they don't say that she imported any more ice, and since they've never had the answer be obtainable only through completely outside information (they've made us go back to other Conundrums, but not calculate outside of what info is given) before AFAIK. I put 667 way back on Thursday, and I'm sticking to my solution.

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Lenny Conundrum
I sent the problem to my brother who graduated from college (this doesn't mean he's right though xp) and this is what he said:

Quote:

Julia, this is a trick question I think.

You would need university calculus or a spreadsheet to figure this out since the volume melted constantly changes as it melts.

However, you can figure out the volumes:

Find the volume of her bathtub to see how much water it takes for it to be full:
area of a hexagon is 6 times the area of an equilateral triangle (all angles are 60 degrees)
The area of a triangle is 1/2 a * b where a is the sides and b is the vertical piece. If you cut the equilateral triangle in half, you will see that you have a 1, 2, root 3 triangle (angles of 30 degrees, 60 degrees and 90 degrees). That means that b will be (root 3)/ 2 times as long as a. Therefore, area of a hexagon is 6 * 1/2 * a * (root3)/2 * a
That equals 3*(root3)/2*a^2. or 2.598 * a * a.
The volume of a hexagon is just the height times the area, so 3*(root3)/2*a^2*h. Or to make is easier, 2.598*a^2*h.
For the bathtub, the volume to fill it would be 2.598*(1.5m)*(1.5m)*(0.8m) or 4.6764m^3.

Now, calculate the volume of the sphere of ice. The formula for volume of a sphere is 4/3*pi*r^3 where r is the radius. The radius is half the diameter, or in this problem 1m.
The volume of the whole sphere of ice is 4/3*pi*1^3 or 4.1888 m^3.
That means that even if the whole sphere of ice melted, the bathtube would not fill up! So, it would take infinity amount of time to fill up the bathtub, because she wouldn't have enough water.

Dave

But I put 667 as the answer on one of my accounts . . .

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Quote:

You would need university calculus or a spreadsheet to figure this out since the volume melted constantly changes as it melts.

Would it be wrong for me to waste my teachers time with this riddle?? Hey maybe they'll think i'm a extra zeallous student lol!

The initial 250 have passed i bet Sad

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Regarding brother Dave's response:

The way the problem is worded, you don't need calculus to determine how long it takes for the sphere to melt. Yes, the volume that melts per minute changes (decreases) over time, but that is irrelevant since the radius changes at a constant rate.

"But I put 667 as the answer on one of my accounts . . ." Shocked

How many accounts do you have?

"Would it be wrong for me to waste my teachers time with this riddle??" I wouldn't bother. It boils down to two choices:
- 667
- never (infinity)
depending on what you think was intended.

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markr wrote:

"But I put 667 as the answer on one of my accounts . . ." How many accounts do you have?

I have five Smile

for storage purposes, since each account besides the creators' can only have four pets.

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