Growing, Growing, Gone
Minus wanted to strut his stuff this week and insisted we post his lengthy solution. We couldn't say no to the big guy because...hey, this is his show.
In the first 2 years, the Kudzu Pine grows
6 + 1.5 = 7.5 meters.
In the first 2 years, the Snailwood Pine grows
5 + 2.5 = 7.5 meters.
They are exactly the same height after two years.
In five years, the Kudzu Pine is
6 + 1.5 + 0.375 + 0.09375 + 0.0234375 = 7.9921875 meters high, or approximately 799 centimeters.
In five years, the Snailwood Pine is
5 + 2.5 + 1.25 + 0.625 + 0.3125 = 9.6875 meters, which is approximately 969 centimeters.
Their difference in height is about 170 centimeters.
As an aside, let's look at what their ultimate height will be.
The formula for a geometric series, which describes these trees' rates of growth, is
T = a0[(1 - rn/(1 - r)]
where T is the total growth, a0 is the amount of growth in the first year, and r is the ratio by which the rate of growth decreases every year. You then set the two amounts equal to each other.
For the Kudzu Pine,
Tk = 6[(1 - 0.25n)/0.75]
which means Tk = 8 - 8(0.25n). Thus, the ultimate limit for the height of the Kudzu Pine is 8 meters.
Tk = 10[1/(1 - 0.25)]
For the Snailwood Pine
Ts = 5[(1 - 0.5n/0.5]
Ts = 10 - 5(0.5n). Thus, the ultimate limit for the Snailwood Pine is 10 meters.
Mrs.William's 5th Graders, at South Scotland School in Laurinburg, NC, were awfully close to the right answer. Minus pointed out that they may have had a round-off error during the cooking process--hence, a slightly taller Snailwood Pine.
We made a chart of each tree and found that the two trees will be equal in height the 2nd year. After 5 years the Snailwood Pine will be 171 cm taller than the Kudzu Pine.
Gordon Caldwell [gcaldwel@MNSi.Net] is this week's numero uno chef. His main-dish solutions look complex, but are actually beautifully executed. Magnifique!
Kudzu: Height = 6+.25(6)+.25^2(6)+.25^3(6)+.....
Using formula for the sum of a geometric series and equating the 2 heights we get the equation:
6[1-(.25)^n]/(1-.25)=5[1-(.5)^n]/(1-.5) where n=# of years
Simplifying we obtain:
(2^n)^2-5(2^n)+4=0 which factors to give
(2^n-4)(2^n-1)=0 from which
2^n-4=0 or 2^n-1=0 which yields
n=2 or n=0
n=0 is an obvious reject.
Therefore, it will take 2 years.
By development of terms.
Therefore it takes 2 years
Esthetically less pleasing but much more efficient, Part B:
After 5 years
H(K)=6+1.5+.375+.09375 + .0234375=7.9921875m
The difference to the nearest cm. is 170cm.
James Morgan, of Clevedon Community School in N. Somerset, UK, provides this week's sweet-treat to round out Minus' meal. James baked up the first part of his solution without a hitch, but the second part was left in the oven for a little too long. Ooops!
The two trees will be the same height after two years. After two years the trees will both be 7.5m tall.