NeoPets Riddles (Lenny Conundrums) and Answers Here

mystery pic - reminds me of something altadorian because of the silver and yellow.

just my impression though xD

I would help with the MP but honestly I am horrible at those, I don't even attempt.

New conundrum! :]

In the Virtupets Space Station, there are two water tanks providing water to the station: a primary water tank, which is a sphere with a 4.5 metre radius, and a much larger backup water tank. During the busy Space Station tourist season, the primary water tank became empty. Dr. Sloth ordered his Grundos to carry bucketfuls of water from the larger backup water tank to the primary water tank.

Dr. Sloth ordered his Grundos to fill the tank so that the water is exactly 6.5 metres deep at the center. If one bucket of water holds exactly 10 litres, how many bucketfuls of water are required to fill the tank to the required depth? Round up to the nearest whole number.

In the Virtupets Space Station, there are two water tanks providing water to the station: a primary water tank, which is a sphere with a 4.5 metre radius, and a much larger backup water tank. During the busy Space Station tourist season, the primary water tank became empty. Dr. Sloth ordered his Grundos to carry bucketfuls of water from the larger backup water tank to the primary water tank.


Dr. Sloth ordered his Grundos to fill the tank so that the water is exactly 6.5 metres deep at the center. If one bucket of water holds exactly 10 litres, how many bucketfuls of water are required to fill the tank to the required depth? Round up to the nearest whole number.


You can find a calculator for the partial volume here: http://www.abe.msstate.edu/~fto/tools/vol/partsphere.html

and remember that 1000 litres= 1 cubic metre


sent in my answere at 6:27

so what is the answer? i am confused!

figure out the volume and multiply it by 100 (1000litres/cubic metre divided by 10 litres per bucket)

volume = 309.708675
309.708675 * 1000 = 309708.675
309708.675/10= 30970.8675
to the nearest whole number: 30971

i think i got it but im not sure, ill send it anyways
thanks again

YAY, i got the same thing as you
x)

wtg bigb

You shouldn't give in so easily.
Should have made him post an answer he got on his own first, at least. =o

For the depth in the primary tank to be 6.5 meters, then the water is two meters past the halfway point. That is, there are h = 2.5 meters of empty space between the top of the water and the top of the sphere.

For any "topped" sphere of radius R, in which the top "h" units in height have been lopped off, the volume formula is as follows:

. . .V = (1/3)(pi)[4R^3 + h^3 - 3Rh^2]

In this case, R = 4.5 and h = 2.5, so the volume is:

. . .V = (1/3)(pi)[364.5 + 15.625 - 84.375]
. . .. .= (1/3)(pi)[295.75]
. .. . .= (295.75/3)(pi)

The units on this volume are "cubic meters", naturally. Since 1 m^3 = 1000 L, then the volume is (295750/3)(pi) L. Since 10 L = 1 "bucket", then the volume is (29575/3)(pi) "buckets". Multiplying this out, we get 30970.867576... "buckets". Rounded to the nearest whole number, this is 30971 "buckets".

. . . .********************
. . . .*****ROUND 165:*****
. . . .********************
. . . .***THE ANSWER IS****
. . . .********************
. . . .*******30971********
. . . .********************

Eliz.

Note: The lopped-off portion of the sphere is called a "spherical cap". Online calculators may be used to compute the above volume, if one views the sphere as being upside-down. That is, instead of subtracting the volume of the "cap" from the total volume of the sphere, instead regard the water-filled portion as being the "cap", and find the volume with sphere radius R = 4.5 and spherical cap height H = 6.5.

round 165
thanks 4 the answer, i hope its 30971

Strange... a sphere 4.5 meters in radius has a volume of

381.7 cubic meters


The volume of the end cap of height H on a sphere radius R is:
(H being the height from the Outside of the sphere.)

vol = ((pi*h*h)/3)*(3*r-h)

plugging in R= 4.5, and H=1.5 (Why 1.5? total height = 4.5*2, or 9 Meters. Subtract the 6.5 meters of water, leaving 1.5 meters of air at the top.)

vol = 28.3 cubic meters

Subtracting the air space from the water, I come up with 353.4 cubic meters.

Yes yes, not enough decimals. I'm simply showing you that the formula in the calculator you used appears to be off...

Double checking my work:

Vsphere = 4/3 Pi (R^3)

Given R=4.5

Vsphere = 381.7035074
-----------------------------
Vcap = 1/3 Pi (H^2) (3R-H)

Given R=4.5, and H=1.5

Vcap = 28.27433388
-----------------------------
Subtracting cap volume from sphere volume gives:

Vsphere - Vcap = 353.4291735
-----------------------------
Converting to liters,

Vsphere = 353429.1735
-----------------------------
Dividing by 10 liter buckets:

Buckets = 35342.91735
-----------------------------
Rounding up to nearest bucket:

35343 buckets
-----------------------------
Good luck on YOUR Lenny!

Stapel's error (and the calculator) may derive from attempting to use an Inside value for a cap.

Or if you prefer, from attempting to calculate a cap which is "taller" than half a sphere.

You can find the identical formulas for Vcap and Vsphere on the page mentioned earlier: http://mathworld.wolfram.com/SphericalCap.html

Or those who know calculus:

Build an integral, calculating the volume of a series of cylinders. Imagine taking cross sections of a sphere. Approximate the section by building a cylinder of the same radius at that cross section. There will be a little "waste" or "excess", but by making our slices thinner and thinner, the error approaches zero.

The radius of each cylinder at height "X" is equal to SQR(r^2 -x^2)

Set your limits from -4.5 to 2.5, and integrate!

Yes - that is exactly the mistake made by those using the calculator.

Try using the SAME calculators to computer the volume of a 1.5 meter cap on a 4.5 meter sphere - you'll get the same volume I did.

Subtract that from the sphere's volume for the same answer.

The Spherical cap formulas used on that site do not work for situations where cap height is greater than the radius.

I hope you didn't submit an answer yet!

um... i think you just made a calculation error there.
9-6.5=2.5
so h should be 2.5 rather than 1.5

You might want to check that:

. . .9 - 6.5 = 2.5

...not 1.5. Sorry.

Eliz.

sphere has a radius r and centered on the origin, the volume in question goes from -r to 2 (6.5 depth) on the x axis.

the radius of the cylinder - (r^2-x^2)^1/2
the area of the cylinder - radius^2*pi=(r^2-x^2)*pi
the height of the cylinder - dx

integrate from -r to 2:
pi*S(r^2-x^2)dx=pi*[r^2*x-1/3*x^3]@(-r,2)= pi*(r^3-1/3*r^3+2r^2-8/3) = 309.7086757663937834251089268683 c.m.
now there are 1000 liters to a cubic meter and 10 liters to a bucket, so there are a 100 buckets to a cubic meter
30970.86757663937834251089268683 rounds up to 30971 bucket