ALG-24

Doubling time

Exponential growth.

Reading speed

Algebradoubling

Governing equation

T_2=\\ln2/k

$k \in \left[0.005, 0.5\right]$

where

k: k (1/yr)
T_2: T2 (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.

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①

Governing law — red

T_2=\\ln2/k

$k \in \left[0.005, 0.5\right]$

$k = 0.005$ $T_2 = 138.6$

Along the curve

②

Domain of the variable — blue

k_{\min} = 0.005,\quad k_{\max} = 0.5

$k \in \left[0.005, 0.5\right]$

$k = 0.005$ $k = 0.005$

Along the curve

③

Response curve — green

T_2=\\ln2/k

$k \in \left[0.005, 0.5\right]$

$k = 0.005$ $T_2 = 138.6$

Along the curve

④

Rate of change — amber

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

$k \in \left[0.005, 0.5\right]$

$k = 0.008$ $T_2^{\prime} = -12348.5$

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-24 — Doubling time) is the form associated with doubling. Working symbols: $k$ $\rightarrow$ $T_2$ . Exponential growth.

Purpose

Purpose: compute

T_2

from

k

in Algebra via

T_2=\\ln2/k

Exponential growth. 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=\\ln2/k

yields

T_2 = 9.90\,\mathrm{yr}

. Exponential growth. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

k k0.07 1/yr

Outputs

T2 T_29.90 yr

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Narration of this film

Exponential growth.

Reading speed

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