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February 8

Problem solving : work backward

How do I solve the problem? In June 2005, a sixth grade class planted a tree in the schoolyard. The tree grew about 3 inched a year. If the tree was 38 inches high in June 2010, about how high was the tree when it was planted? — Preceding unsigned comment added by 67.8.185.89 (talk) 04:16, 8 February 2012 (UTC)[reply]

Start with:

PRESENT HEIGHT = INITIAL HEIGHT + GROWTH

Now subtract GROWTH from both sides:

PRESENT HEIGHT - GROWTH = INITIAL HEIGHT + GROWTH - GROWTH

PRESENT HEIGHT - GROWTH = INITIAL HEIGHT

Now expand GROWTH:

PRESENT HEIGHT - (YEARS OF GROWTH)×(GROWTH PER YEAR) = INITIAL HEIGHT

Now just plug in your values to get the answer. StuRat (talk) 04:50, 8 February 2012 (UTC)[reply]

Time lag in second-order positive feedback

In a second-order positive feedback loop that would produce hyperbolic growth with no time lag, what's the shape of the curve if there's a constant time lag? And what about a time lag that's inversely proportional to the quantity -- does the latter still produce a singularity? Neon Merlin 04:46, 8 February 2012 (UTC)[reply]

I don't have an exact solution, but the double exponential function

e^{2^{t}}

comes close to solving

{\frac {dx}{dt}}={\bigl (}x(t-1){\bigr )}^{2}

. As for the shrinking delay,

x={\frac {-k}{t}}

(where

k\approx 2.1479

solves

(k+1)^{2}=k^{3}

) solves

{\frac {dx}{dt}}=\left[x\left(t-{\frac {1}{x(t)}}\right)\right]^{2}

. --Tardis (talk) 06:32, 16 February 2012 (UTC)[reply]

Value of infinite limit

I'm having trouble with this, the equation of motion from the free fall article (position as a function of time) for a body falling under gravity but subject to air resistance proportional to the square of velocity:

$y=y_{0}-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)$

As t becomes infinite, the equation should become

$y=y_{0}-v_{\infty }{t}$

but I get a -ln2 term appearing. What's wrong?86.174.199.35 (talk) 17:19, 8 February 2012 (UTC)[reply]

It doesn't start going at rerminal velocity immediately so you need y0 adjusted by some constant amount. Dmcq (talk) 17:45, 8 February 2012 (UTC)[reply]

Of course. 86.174.199.35 (talk) 08:28, 9 February 2012 (UTC)[reply]

Actually thinking about it one can have an amount that grew to infinity if it did it slowly enough but that's not what's happening in this case thankfully :) Dmcq (talk) 13:06, 9 February 2012 (UTC)[reply]