The 'Lenny' authors, in Round 245, wrote:On Neopia's first New Year's Eve, some of Neopia's citizens planted a pine tree in the Haunted Woods to commemorate the first New Year's celebration. The tree was just a sapling; the trunk was cone-shaped, and its circumference at the base was 14 cm and its height was 124 cm.
Every year since then, its height has increased 45% over the previous year, and its circumference has increased 22%. If the density of the wood is 530 kilogrammes per cubic metre, how many kilogrammes did the tree's trunk weigh this past New Year's Eve?
This is compounded growth, since each year's growth is increased "over the previous year". So use the compounded growth formula:
. . . . .A = P(1 + r/n)^(nt)
...where:
. . . . .A: ending amount
. . . . .P: beginning amount
. . . . .r: growth rate (as a decimal)
. . . . .n: number of compoundings per period
. . . . .t: number of periods
In this case:
. . . . .P: the original value for height h or
. . . . . . .circumference C
. . . . .r: 0.45 (for h) or 0.22 (for C)
. . . . .n: n = 1 for yearly compounding
. . . . .t: unknown (to residents of the real world;
. . . . . . .Neopians will need to supply this value)
In "t" years, the height h will have gone from an initial value of h_0 = 124 to a final value given by:
. . . . .h(t) = 124(1 + 0.45)^(t)
. . . . . .. . .= 124(1.45)^(t)
In "t" years, the circumference C will have gone from an intial value of C_0 = 14 to a final value given by:
. . . . .C(t) = 14(1 + 0.22)^(t)
. . . . . .. . .= 14(1.22)^(t)
The formula for the volume V of a right circular cone with radius r and height h is given by:
. . . . .V = (1/3)(pi)(r^2)(h)
The circumference C of a circle with radius r is given by:
. . . . .C = 2(pi)r
Then the volume, in terms of the circumference (rather than the radius) and the height, is given by:
. . . . .C = 2(pi)r
. . . . .C / (2(pi)) = r
. . . . .V = (1/3)(pi)(r^2)(h)
. . . .. . .= (1/3)(pi)[C / (2(pi))]^2 (h)
. . .. . . .= (1/3)(pi)[C^2 / (4(pi)^2)](h)
. .. . . . .= (1/3)(pi)(C^2)(h) / (4)(pi)^2
.. . . . . .= (1/3)(1/4)(pi / (pi)^2)(C^2)(h)
. . . . . ..= (1/12)(1/pi)(C^2)(h)
Then the volume V of the cone, after t years, is:
. . . . .V = (1/12)(1/pi)[14(1.22)^t)]^2 [124(1.45)^t)]
. . . . .. .= (1/12)(1/pi)(196)[(1.22)^(2t)](124)[(1.45)^t]
. . . . .. .= (6076/3pi)(1.22)^(2t) (1.45)^(t)
This is, of course, the volume in terms of cubic centimeters. The density D of the wood is given in terms of cubic meters: 530 kilograms per one cubit meter. Since there are one hundred centimeters to one meter, then there are 100^3 = 1 000 000 cubic centimeters to one cubic meter.
To determine the final required value (the mass of wood), first obtain the value of t, being the number of years. Plug this into the volume formula (above) to obtain the value of the volume V of wood in cubic centimeters. Divide this volume value by 1 000 000 to obtain the volume stated in terms of the number of cubic meters. Then multipy this new volume value by 530 to obtain the number of kilograms of wood.
Note: I see no instructions regarding rounding.
If the first New Year's Eve fell on 31 December 1999, then 31 December 2007 gives t = 8 (since 31 December 2000 would have been t = 1). Plugging this into the algorithm above, one should obtain a value of 160.81398577... kilograms.
Rounded in the standard (mathematical) manner, one would obtain an answer of "161 kg". I do not know whether the units (namely, "kilograms") should be included within the submitted solution.
Please check my work.
Eliz.
. So with that change I get about 29.4 kg.
