Question

suppose that the rate at which body weight w changes with age t is \\( \frac{dw}{dt} \propto w^{a} \\), where a > 0 is a coefficient that takes different values for different species of animal. (a) the relative growth rate (percentage weight gained per unit of time) is defined as \\( g(w)=\frac{1}{w} \frac{dw}{dt} \\). write down a formula for g(w). for which values of a is the relative growth rate increasing, and for which values is it decreasing? (b) as fish grow larger, their weight increases each day but the relative growth rate decreases. if the rate of growth is described by \\( \frac{dw}{dt} \propto w^{a} \\), explain what constraints must be imposed on a. (a) write down a formula for g(w). let k be a constant of proportionality where k > 0. \\( g(w)=k(\square) \\)

Explanation:

Step1: Substitute $\frac{dW}{dt}=kW^{a}$ into $G(W)$ formula

Given $G(W)=\frac{1}{W}\frac{dW}{dt}$ and $\frac{dW}{dt}\propto W^{a}$, so $\frac{dW}{dt} = kW^{a}$ (where $k>0$ is the constant of proportionality). Then $G(W)=\frac{1}{W}\times kW^{a}$.

Step2: Simplify the expression

Using the rule of exponents $x^{m}\times x^{n}=x^{m + n}$ and $\frac{x^{m}}{x^{n}}=x^{m - n}$, we have $G(W)=kW^{a - 1}$.

Step3: Find when $G(W)$ is increasing or decreasing

Differentiate $G(W)$ with respect to $W$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we get $G^\prime(W)=k(a - 1)W^{a - 2}$.

• For $G(W)$ to be increasing:

We need $G^\prime(W)>0$. Since $k>0$ and $W>0$ (weight is non - negative), then $(a - 1)W^{a - 2}>0$. Since $W^{a - 2}>0$ for $W>0$, we need $a-1>0$, so $a > 1$.

• For $G(W)$ to be decreasing:

We need $G^\prime(W)<0$. Since $k>0$ and $W>0$, then $(a - 1)W^{a - 2}<0$. Since $W^{a - 2}>0$ for $W>0$, we need $a - 1<0$, so $0 < a<1$.

Answer:

$G(W)=kW^{a - 1}$; $G(W)$ is increasing when $a>1$ and decreasing when $0 < a<1$.