<Text-field style="Heading 1" layout="Heading 1">Dimensional and Dimensionless Analysis</Text-field>

In the SI system, there are 7 fundamental dimensional units:

mass, length, time, current, temperature, amount of substance, and luminous intensity. For our example, we will only be concerned with mass, length, and time, and these will represent our fundamental dimensional units.

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print(); # input placeholder

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The remainder of our descriptive quantities in our model have dimensions that are products of powers of our fundamental dimensional units.

time

length

density

oven temperature

thermal conductivity

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print(); # input placeholder

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(Here temperature is defined as energy per volume, and thermal conductivity is energy times the length divided by the product area, the time, and the temperature)

<Text-field style="Heading 2" layout="Heading 2">Using T as the secondary quantity</Text-field>

Since there are three fundamental dimensional units, we will have three primary quantities which can be chosen arbitrarily from our four variables. We will choose l, \317\201, and \316\272 as our primary variables, giving us three linearly independent vectors with respect our fundamental dimensional units LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiRRIjFGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y8LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==, LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y8LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==, and LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2Ji1JI21uR0YkNiRRIjNGJy9GM1Enbm9ybWFsRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y8LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw== :

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JSFH

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

JSFH

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

JSFH

We will now create our primary quantity D, such that [D]=[t], where 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy9GNlEnbm9ybWFsRic= are linearly independent primary quantities. To do this, we need to solve 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. First, we will construct the augmented matrix:JSFH

print(); # input placeholder

NiI=

JSFH

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKmMlKVxeIg==

Solving the above matrix yields:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FALUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKmMlKVxeIkYnLyUrYWN0aW9udHlwZUdROm1hcGxlc29mdC5jb206bGFiZWwoTDE0MSlGJy8lJXZpZXdHUSZsYWJlbEYnRjovJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGTUZKRk0=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKiVlKVxeIg==

From this, we can solve for our variables \316\261, \316\262, and \316\263:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyUvSSZhbHBoYUc2IiIiIy9JJWJldGFHRiUiIiEvSSZnYW1tYUclKnByb3RlY3RlZEchIiI=

JSFH

Therefore, 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, which has the same dimensions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Next, we will calculate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjFGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== based on our secondary quantity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, such that [D]=[T]. To do this, we solve 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. First, we construct the augmented matrix:

JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShjb252ZXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjs2Ji1GIzYrLUYsNiVRImxGJ0Y0RjctSSNtb0dGJDYtUSIsRicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRk4vJSpzeW1tZXRyaWNHRk4vJShsYXJnZW9wR0ZOLyUubW92YWJsZWxpbWl0c0dGTi8lJ2FjY2VudEdGTi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRJHJob0YnL0Y1Rk5GSkZGLUZHNi1RIn5GJ0ZKRkwvRlBGTkZRRlNGVUZXRllGZW4vRmluRmduLUYsNiVRJmthcHBhRidGXm9GSkZGLUY7NiYtRiM2Ki1JI21uR0YkNiRRIjFGJ0ZKRkYtRkc2LVEqJnVtaW51czA7RidGSkZMRmJvRlFGU0ZVRldGWS9GZm5RLDAuMjIyMjIyMmVtRicvRmluRmNwRltwRkZGX3AtRlxwNiRRIjJGJ0ZKRkpGSi8lJW9wZW5HUScmbGFuZztGJy8lJmNsb3NlR1EnJnJhbmc7RidGSkZKL0ZpcFEiW0YnL0ZccVEiXUYnRkZGX28tRiw2JVEnTWF0cml4RidGNEY3RkpGSkZKRisvJStleGVjdXRhYmxlR0ZORko=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKlsnKVxeIg==

Solving the matrix yields:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUTZSZWR1Y2VkUm93RWNoZWxvbkZvcm1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JkYrLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRQC1JJ01hdHJpeEc2IjYjL0kkJWlkR0YkIipbJylcXiJGJy8lK2FjdGlvbnR5cGVHUTptYXBsZXNvZnQuY29tOmxhYmVsKEwyNzQpRicvJSV2aWV3R1EmbGFiZWxGJ0YrL0Y4USdub3JtYWxGJ0ZMRkxGKy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRkw=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKjcoKVxeIg==

From this, we can solve for our variables \316\261, \316\262, and \316\263:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyUvSSZhbHBoYUc2IiEiIy9JJWJldGFHRiUiIiIvSSZnYW1tYUclKnByb3RlY3RlZEciIiM=

JSFH

Therefore, 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, which has the same dimensions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.JSFH

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

To create our dimensionless quantities LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnRjUvRjNRJXRydWVGJy9GNlEnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjU=, we can simply divide our primary and secondary quantities by our newly created quantities LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjFGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0Y1 . This give us:

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JSFH

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

Applying the Buckingham Pi Theorem, we obtain the dependency 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, with an unknown function F. In order to gain further information on F, we will need to invoke experimental data. Before doing that, we will simplify our equation by making a few assumptions. Since we are testing the validity of the rule of thumb, we will first assume that we'll keep the oven temperature constant at 350 \302\260F. Also, it would be reasonable to assume that the thermal conductivity and density of turkeys does not vary much from one bird to the next. The final assumption we'll make is that the mass of a turkey (M) is proportional to the volume of the turkey. Since volume is in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4= and mass is in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEja2dGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn, we can make the substitution 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. Making the substitution will give us 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. This relationship is rather useless, since all it shows is that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Looking at the equation, we know that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is dimensionless, and temperature, density and thermal conductivity are kept fixed. Only the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= remains, and we have to proceed by testing simple hypotheses for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

The simplest hypotheses is that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is constant. If this is the case, than cooking time is proportional to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjJGJy9GNlEnbm9ybWFsRictRiM2JS1GPjYkUSIzRidGQUYyRjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRk0vJSliZXZlbGxlZEdRJmZhbHNlRidGMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZBas opposed to being a linear function as stated by the rule of thumb. However, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1cEdGJDYlLUYsNiVRIk1GJ0YvRjItRiM2JS1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLUYjNiUtRkY2JFEiM0YnRklGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZVLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGSUZJRkk= is not known to be constant. If, for example 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 where C is some constant, than the resulting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= dependence on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= would in fact be linear.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Using \316\272 as the secondary quantity</Font></Text-field>

Repeating a similar process as above, we will choose l, \317\201, and T as our primary variables:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

We will now solve 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. First, we will construct the augmented matrix:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

print(); # input placeholder

NiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoY29udmVydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYoLUY2NiYtRiM2LC1GLDYlUSJsRidGL0YyLUkjbW9HRiQ2LVEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUScmIzk2MTtGJy9GMEZJRkVGQS1GQjYtUSJ+RidGRUZHL0ZLRklGTEZORlBGUkZURlYvRlpGWC1GLDYlUSJURidGL0YyRkEtRjY2Ji1GIzYpLUkjbW5HRiQ2JFEiMEYnRkVGQUZmb0ZBLUZnbzYkUSIxRidGRS8lK2V4ZWN1dGFibGVHRklGRUZFLyUlb3BlbkdRJyZsYW5nO0YnLyUmY2xvc2VHUScmcmFuZztGJ0ZdcEZFRkUvRmBwUSJbRicvRmNwUSJdRidGQUZqbi1GLDYlUSdNYXRyaXhGJ0YvRjJGXXBGRUZFRl1wRkU=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKncoKVxeIg==

Solving the above matrix yields:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1FALUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKncoKVxeIkYnLyUrYWN0aW9udHlwZUdROm1hcGxlc29mdC5jb206bGFiZWwoTDM1MClGJy8lJXZpZXdHUSZsYWJlbEYnRjovJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGTUZKRk0=

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKmtjM20i

From this, we can solve for our variables \316\261, \316\262, and \316\263:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyUvSSZhbHBoYUc2IiIiIi9JJWJldGFHRiUjRiYiIiMvSSZnYW1tYUclKnByb3RlY3RlZEcjISIiRio=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Therefore, 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, which has the same dimensions of . Next, we will calculate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjFGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== based on our secondary quantity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, such that [D]=[\316\272]. To do this, we solve 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. First, we construct the augmented matrix:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKj9iM20i

Solving the matrix yields:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUTZSZWR1Y2VkUm93RWNoZWxvbkZvcm1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JkYrLUkobWFjdGlvbkdGJDYlLUkqbXZlcmJhdGltR0YkNiNRPy1JJ01hdHJpeEc2IjYjL0kkJWlkR0YkIio/YjNtIkYnLyUrYWN0aW9udHlwZUdROm1hcGxlc29mdC5jb206bGFiZWwoTDM2MClGJy8lJXZpZXdHUSZsYWJlbEYnRisvRjhRJ25vcm1hbEYnRkxGTEYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGTA==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKjchPmc7

From this, we can solve for our variables \316\261, \316\262, and \316\263:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyUvSSZhbHBoYUc2IiIiIi9JJWJldGFHRiUjISIiIiIjL0kmZ2FtbWFHJSpwcm90ZWN0ZWRHI0YmRis=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Therefore,

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

, which has the same dimensions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NTQ7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

Applying the Buckingham Pi Theorem, we obtain the dependency 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. Making the same assumptions about temperature, thermal conductivity and mass, we will get 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.

JSFH

<Text-field style="Heading 2" layout="Heading 2">Using mass instead of density and length</Text-field>

A third option is to use T, \316\272, and mass M. This will give us:

JSFH

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

JSFH

We will now solve 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. First, we will construct the augmented matrix:JSFH

print(); # input placeholder

NiI=

JSFH

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKmdfUm4i

Solving the above matrix yields:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE2UmVkdWNlZFJvd0VjaGVsb25Gb3JtRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2I1EhRictSShtYWN0aW9uR0YkNiUtSSptdmVyYmF0aW1HRiQ2I1E/LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKmdfUm4iRicvJSthY3Rpb250eXBlR1E6bWFwbGVzb2Z0LmNvbTpsYWJlbChMMzkzKUYnLyUldmlld0dRJmxhYmVsRidGOi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZNRkpGTQ==

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

LUknTWF0cml4RzYiNiMvSSQlaWRHRiQiKndIYm8i

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

From this, we can solve for our variables \316\261, \316\262, and \316\263:

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

LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic=

print(); # input placeholder

NyUvSSZhbHBoYUc2IiMhIiMiIiYvSSViZXRhR0YlIyEiIkYoL0kmZ2FtbWFHJSpwcm90ZWN0ZWRHIyIiI0Yo

JSFH

Therefore, 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. Since there is no secondary quantity, we can simply create our dimensionless quantity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5Mjg7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy as 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. Making the same assumptions about temperature, thermal conductivity and mass, we will get 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.

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

We now have three functions to describe the cooking time of a turkey in addition to the rule of thumb. In order to determine which option yields the best results, we will need to analyze some data. The following chart has actual cooking times for various sizes of turkeys:

weight (lbs)

time (hrs)

2.5

3.4

3.5

4.5

5.4

5.8

JSFH

Because we do not have any information regarding the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1cEdGJDYlLUYsNiVRIk1GJ0YvRjItRiM2JS1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLUYjNiUtRkY2JFEiM0YnRklGL0YyLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZVLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRi9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGSUZJLUYsNiNRIUYnRkk=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiw2JVEiR0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUklbXN1cEdGJDYlLUYsNiVRIk1GJ0YyRjUtRiM2JC1JJm1mcmFjR0YkNigtSSNtbkdGJDYkUSIxRicvRjZRJ25vcm1hbEYnLUYjNiQtRkk2JFEiM0YnRkxGTC8lLmxpbmV0aGlja25lc3NHRksvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGVy8lKWJldmVsbGVkR1EmZmFsc2VGJ0ZMLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZMRkxGTA==, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiSEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj5GPg==, we will just test the models under the assumptions that these functions are constant. We will now plot the dependence of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRInRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1JJW1zdXBHRiQ2JS1GLzYlUSJNRidGMkY1LUYjNiUtRi82JVEicEYnRjJGNUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGTS8lKWJldmVsbGVkR1EmZmFsc2VGJy9GNlEnbm9ybWFsRic= against the weight LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= where p=2/3, 2/5, 1/3 respectively, along with a line of best fit.

JSFH

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

print(); # input placeholder

JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRJ1ZlY3RvckYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiOEYnRjkvJStleGVjdXRhYmxlR0Y9LyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnRjlGOS1GNjYtUSI6RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUZlbkZnbkY5

print(); # input placeholder

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

JSFH

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

print(); # input placeholder

JSFHJSFH

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print(); # input placeholder

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JSFH

As we can see, all three can be represented by a linear function, but only p=2/3 is essentially constant. Therefore the cooking time would be best represented by the function 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. Simplifying this will give us 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is some constant