Question

allometric equations describe the scaling relationship between two measurements such as skull length versus body length. in vertebrates, we typically find that skull length ∝ body length^a for 0 < a < 1. express the growth rate of the skull length in terms of the growth rate of the body length.
( type an expression using s and b as the variables. )
\frac{ds}{dt}=\frac{\square}{dt}

Explanation:

Step1: Use the chain rule

The chain rule states that if \(y = f(u)\) and \(u = g(x)\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). In this case, \(S\) is a function of \(B\) (since \(S\propto B^{a}\), or \(S = k\cdot B^{a}\) for some constant \(k\)), and \(B\) is a function of \(t\). So, \(\frac{dS}{dt}=\frac{dS}{dB}\cdot\frac{dB}{dt}\).

Step2: Solve for \(\frac{dS}{dt}\)

We want to express \(\frac{dS}{dt}\) in terms of \(\frac{dB}{dt}\). Rearranging the chain - rule formula \(\frac{dS}{dt}=\frac{dS}{dB}\cdot\frac{dB}{dt}\) gives \(\frac{dS}{dt}\div\frac{dB}{dt}=\frac{dS}{dB}\), or \(\frac{dS}{dt}= \frac{dS}{dB}\cdot\frac{dB}{dt}\).

Answer:

\(\frac{dS}{dt}\)