Question

suppose that the rate at which body weight w changes with age t is \\( \frac { d w } { d t } \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 { d w } { d t } \\). 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 { d w } { d t } \propto w ^ { a } \\), explain what constraints must be imposed on a. a. the relative growth is decreasing for \\( a > \\). (type an integer or a simplified fraction.) b. the relative growth is decreasing for \\( 0 < a < 1 \\). (type an integer or a simplified fraction.) c. the relative growth is not decreasing for any values of a. (b) in order for the relative growth rate to decrease as fish grow larger, the value of a has to be

Explanation:

Step1: Find formula for \(G(W)\)

Since \(\frac{dW}{dt}\propto W^{a}\), we can write \(\frac{dW}{dt} = kW^{a}\) (where \(k>0\) is a constant of proportionality).
Given \(G(W)=\frac{1}{W}\frac{dW}{dt}\), substitute \(\frac{dW}{dt} = kW^{a}\) into the formula for \(G(W)\):
\(G(W)=\frac{1}{W}\times kW^{a}=kW^{a - 1}\).

Step2: Determine when \(G(W)\) is increasing or decreasing

Find the derivative of \(G(W)\) with respect to \(W\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).
\(G^\prime(W)=k(a - 1)W^{a - 2}\).
Since \(k>0\) and \(W>0\) (weight is non - negative), the sign of \(G^\prime(W)\) is determined by \((a - 1)\).

• If \(a-1>0\) (i.e., \(a > 1\)), then \(G^\prime(W)>0\) and \(G(W)\) is increasing.
• If \(a - 1<0\) (i.e., \(0 < a<1\)), then \(G^\prime(W)<0\) and \(G(W)\) is decreasing.

Step3: Solve part (b)

We know that for fish, as \(W\) increases (\(W>0\)), \(G(W)\) decreases.
From \(G(W)=kW^{a - 1}\), for \(G(W)\) to be decreasing as \(W\) increases, we need \(G^\prime(W)<0\).
Since \(G^\prime(W)=k(a - 1)W^{a - 2}\), \(k>0\) and \(W>0\), we must have \(a-1<0\) (because \(W^{a - 2}>0\) for \(W>0\) and \(a>0\)). So \(0 < a<1\).

Answer:

(a) \(G(W)=kW^{a - 1}\), \(G(W)\) is increasing for \(a > 1\) and decreasing for \(0 < a<1\).
(b) \(0 < a<1\)