<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

<Text-field style="Heading 1" layout="Heading 1">Mathematical model</Text-field>

We model our partially burned candle as a truncated cone (or frustrum for all you Lateinophiles)

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Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== = Height of the candle t hours after it is lit

Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== = Radius of the top of the candle at time t.

Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= = Height of the candle when it is new

Let R = Radius of the bottom of the candle

Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiVkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== = Volume of the candle at time t

Let k = the constant rate at which wax is consumed

Distances are in centimeters, times are in hours, and

volumes are in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUYjNiUtSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GMTYlUSJtRidGNEY3RjctRiM2JS1JI21uR0YkNiRRIjNGJ0Y3L0Y1USV0cnVlRicvRjhRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIi5GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGaG5GNw==

<Text-field style="Heading 1" layout="Heading 1"></Text-field>

Note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUYsNiVRIkhGJ0YvRjJGPg==, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tSSNtb0dGJDYtUSI9RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0ZGLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIzAuRidGPkY+

The volume of the frustrum is the volume of the new candle minus the volume that has already burned. This is given by

JSFH

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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

Ly1JIlZHNiI2I0kidEdGJSwmKiojIiIiIiIkRitJI1BpRyUqcHJvdGVjdGVkR0YrKUkiUkdGJSIiI0YrSSJIR0YlRitGKyoqRipGK0YtRispLUkickdGJUYmRjFGKywmRjJGKy1JImhHRiVGJiEiIkYrRjo=

JSFH

We can eliminate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== using the following proportion, which holds by similar triangles.

JSFH

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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

LyomSSJSRzYiIiIiLUkickdGJTYjSSJ0R0YlISIiKiZJIkhHRiVGJiwmRi1GJi1JImhHRiVGKUYrRis=

JSFH

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVExaXNvbGF0ZX5mb3J+cih0KUYnRi9GMkZD

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

Ly1JInJHNiI2I0kidEdGJSooLCZJIkhHRiUiIiItSSJoR0YlRiYhIiJGK0kiUkdGJUYrRipGLg==

Substituting into , we get

JSFH

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

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

Ly1JIlZHNiI2I0kidEdGJSwmKiojIiIiIiIkRitJI1BpRyUqcHJvdGVjdGVkR0YrKUkiUkdGJSIiI0YrSSJIR0YlRitGKyosRipGK0YtRispLCZGMkYrLUkiaEdGJUYmISIiRixGK0YvRispRjJGMUY4Rjg=

But since we assumed the candle loses volume at a constant rate k, the second term above is equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtaUdGJDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUYsNi1RIn5GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRJjAuMGVtRicvRkVGUy1GRzYlUSJ0RidGSkZNRi8=.

Thus, we have the following volume equation, which we can then solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj0tSSNtb0dGJDYtUSIuRidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRS8lKXN0cmV0Y2h5R0ZFLyUqc3ltbWV0cmljR0ZFLyUobGFyZ2VvcEdGRS8lLm1vdmFibGVsaW1pdHNHRkUvJSdhY2NlbnRHRkUvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZURj0=

LywkKipJI1BpRyUqcHJvdGVjdGVkRyIiIiwmSSJIRzYiRictSSJoR0YqNiNJInRHRiohIiIiIiRJIlJHRioiIiNGKSEiIyNGJ0YwKiZGLkYnSSJrR0YqRic=

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

LUkoaXNvbGF0ZUc2IjYkLywkKipJI1BpRyUqcHJvdGVjdGVkRyIiIiwmSSJIR0YkRistSSJoR0YkNiNJInRHRiQhIiIiIiRJIlJHRiQiIiNGLSEiIyNGK0YzKiZGMUYrSSJrR0YkRitGLg==

Ly1JImhHNiI2I0kidEdGJSwmKiYiIiQjIiIiRioqLEYnRixJImtHRiVGLEkjUGlHJSpwcm90ZWN0ZWRHISIiSSJSR0YlISIjSSJIR0YlIiIjRitGMUY0Riw=

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

LWYqNiNJImdHNiJGJkYmRiZDJC1JJ2Fzc2lnbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjLy1JI29wR0YrNiQiIiEtSSRsaHNHRis2IzkkLUkodW5hcHBseUdGJjYkLUkkcmhzR0YrRjUtRjA2I0YzRi9GJkYmRiY2Iy4vLUkiaEdGJjYjSSJ0R0YmLCYqJiIiJCMiIiJGRyosRkRGSUkia0dGJkZJSSNQaUdGKyEiIkkiUkdGJiEiI0kiSEdGJiIiI0ZIRk1GUEZJ

SSJoRzYi

JSFH

Thus, the height of the candle shrinks in proportion to the cube root of time.

Let's also examine the rate of shrinking.

LUklZGlmZkclKnByb3RlY3RlZEc2JC1JImhHNiI2I0kidEdGKEYq

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LCQqLiIiJCMiIiJGJCosSSJ0RzYiRiZJImtHRilGJkkjUGlHJSpwcm90ZWN0ZWRHISIiSSJSR0YpISIjSSJIR0YpIiIjI0YvRiRGKkYmRitGLUYuRi9GMEYxI0YtRiQ=

LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNj9RJyZyYXJyO0YnLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRicvJSVzaXplR1EjMTJGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR0Y3LyUqdW5kZXJsaW5lR0Y3LyUqc3Vic2NyaXB0R0Y3LyUsc3VwZXJzY3JpcHRHRjcvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GJy8lJ29wYXF1ZUdGNy8lK2V4ZWN1dGFibGVHRjcvJSlyZWFkb25seUdGNy8lKWNvbXBvc2VkR0Y3LyUqY29udmVydGVkR0Y3LyUraW1zZWxlY3RlZEdGNy8lLHBsYWNlaG9sZGVyR0Y3LyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmdvLUkmbXRleHRHRiQ2NlEyYXNzdW1pbmd+cG9zaXRpdmVGJ0YvRjJGNUY4RjpGPEY+RkBGQ0ZGRkhGSkZMRk5GUEZSRlRGVkZZRmNv

LUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwkKi4iIiQjIiIiRisqLEkidEdGJ0YtSSJrR0YnRi1JI1BpR0YlISIiSSJSR0YnISIjSSJIR0YnIiIjI0Y0RitGMEYtRjFGMkYzRjRGNUY2I0YyRisuL0knYXNzdW1lR0YkSSlwb3NpdGl2ZUdGJQ==

LCQqLiIiJCMiIiJGJEkjUGlHJSpwcm90ZWN0ZWRHIyEiIkYkSSJ0RzYiIyEiI0YkSSJrR0YsRiVJIlJHRixGLUkiSEdGLCMiIiNGJEYp

JSFH

This expression makes intuitive sense. First of all, it's negative, meaning the height never increases. More importantly, it predicts that the taller and thinner the candle LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiQtSSZtZnJhY0dGJDYoLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYxNigtSSNtbkdGJDYkUSIyRicvRj5RJ25vcm1hbEYnLUYjNiUtRkU2JFEiM0YnRkhGOkY9LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZULyUpYmV2ZWxsZWRHUSZmYWxzZUYnRjpGPS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRiM2JS1GNDYlLUY3NiVRIlJGJ0Y6Rj1GQEZaRjpGPUZPRlJGVUZXRkhGSEZI, the faster its height shrinks with time. This is what every kid knows.

<Text-field style="Heading 1" layout="Heading 1">Visualization</Text-field>

Notice that for a fixed height and burn rate, decreasing the diameter makes the graph steeper, meaning the candle's height will shrink more quickly.

Initial height (cm) updateGraph():

Diameter at the bottom (cm) updateGraph():

Burn rate (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JJW1zdXBHRiQ2JS1JI21pR0YkNiVRI2NtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYkUSIzRicvRjlRJ25vcm1hbEYnRjVGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRiM2JS1GMjYlUSVob3VyRidGNUY4RjVGOC8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGUC8lKWJldmVsbGVkR1EmZmFsc2VGJ0ZB)

updateGraph():

JSFH

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JSFH

updateGraph:=proc() local h, k, H, R, tMax, Hmax: use DocumentTools in h:= -3^(1/3)*(t*k*H^2/(Pi*R^2))^(1/3)+H: H:=Do(%H): Hmax := 1.2 * H: R:=max(0.1, Do(%diameter)/2): k:=Do(%k): Do(%candlePlot=plot(h, t=0..40, y=0..Hmax, labels=["t (hours)","Height (cm)"], title="Height of the Candle vs. Time", gridlines=true, labeldirections=[horizontal,vertical])): end use: end proc:

JSFH

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