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I recently asked a question to find a formula for a fiction project. I was met with a brilliant answer; but, I am missing key pieces of information to arrive at how the answer is wholly implemented.

Fiction Author Searching For Formula.

I would like to know how the constant representing the mouse (0.259) or wolf (5.021) was reached-so I would be able to alter a constant (likely the wolf) using a different animal, as to change the degree of scaling.

As an example, if we used a 4ft mouse and 12ft elephant as constants; how do I arrive at the 0.259 (mouse) and whatever number the elephant would be? What was the mathematical process?

This way I would be able to convert any two animals into constants and have more control of the scale.

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Lorem Ipsum
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    Tell us the weights of the mouse and the elephant, and then we can solve a system of equations (two equations with two unknowns) to figure out what $a$ and $b$ should be in your formula to make the heights come out as $4$ for the mouse and $12$ for the elephant. – littleO Oct 22 '16 at 19:41
  • The information is in the linked post.

    The mouse weighs 0.019kg and the elephant weighs 7,000kg.

     – Lorem Ipsum Oct 22 '16 at 19:56
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    If I understand correctly, the formula you want to use is $\text{height} = a + b \log(\text{weight})$. Plugging in the height and weight of the mouse, we find that $4 = a + b \log(.019)$. Plugging in the height and weight of the elephant, we find that $12 = a + b \log(7000)$. So we have two linear equations for the two unknowns $a$ and $b$. We can solve for $a$ and $b$ using algebra. – littleO Oct 22 '16 at 20:09
  • Can you walk me through solving them? Please. I understand we need to pick a variable to isolate in each-correct? Then, substitute one solution as an unknown. Finally, we isolate once more? – Lorem Ipsum Oct 22 '16 at 22:22

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If I understand correctly, you want to determine heights using the formula $$ \text{height} = a + b \log(\text{weight}). $$ Plugging in the height and weight of the mouse, we find that $$ \tag{$\heartsuit$} 4 = a + b \log(.019). $$ Plugging in the height and weight of the elephant, we find that $$ \tag{$\spadesuit$}12 = a + b \log(7000). $$ Now let's use equation ($\heartsuit$) to solve for $a$. Subtracting $b \log(.019)$ from both sides of the equation, we find that $$ \tag{$\clubsuit$}a = 4 - b \log(.019). $$ Next, let's plug this value for $a$ into equation ($\spadesuit$). This gives us $$ 12 = 4 - b \log(.019) + b \log(7000). $$ Solving for $b$, we find that $$ b = \frac{-8}{\log(.019) - \log(7000)} \approx .62417. $$ Finally, plug this value for $b$ into equation ($\clubsuit$) to get $a$.

littleO
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  • I think we solved it. The unknown a ≈ 6.474 and b ≈ 0.624! This plugged into the original formula gives both those heights.

    Thank you, so very much! :)

     – Lorem Ipsum Oct 23 '16 at 17:59