ALG-13

Doubling time

T2 = ln 2 / k for y = y0 e^{kt}. Time to double.

Reading speed

ExponentialDoubling time

Governing equation

T_2=\ln 2/k

$k \in \left[0.001, 2\right]$

where

k: Growth rate (1/yr)
T_2: Doubling time (yr)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

T_2=\ln 2/k

$k \in \left[0.001, 2\right]$

$k = 0.001$ $T_2 = 693.1$

Along the curve

②

Domain of the variable — blue

k_{\min} = 0.001,\quad k_{\max} = 2

$k \in \left[0.001, 2\right]$

$k = 0.001$ $k = 0.001$

Along the curve

③

Response curve — green

T_2=\ln 2/k

$k \in \left[0.001, 2\right]$

$k = 0.001$ $T_2 = 693.1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}T_2}{\mathrm{d}k}

$k \in \left[0.001, 2\right]$

$k = 0.014$ $T_2^{\prime} = -26512$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-13 — Doubling time) is the form associated with Doubling time. Working symbols: $k$ $\rightarrow$ $T_2$ . Set e^{k T2} = 2. The Rule of 70 is 70 / (100k) in percent-growth units.

Purpose

Purpose: compute

T_2

from

k

in Algebra via

T_2=\ln 2/k

T2 = ln 2 / k for y = y0 e^{kt}. Time to double. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k = 0.070\,\mathrm{1/yr}

, the governing relation

T_2=\ln 2/k

yields

T_2 = 9.902\,\mathrm{yr}

. Continuous compounding, k in 1/time. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Growth rate k0.07 1/yr

Outputs

Doubling time T_29.902 yr

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Continuous compounding, k in 1/time.

Set e^{k T2} = 2. The Rule of 70 is 70 / (100k) in percent-growth units.

Reading speed

Watch on YouTube