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.
for which values of a is the relative growth rate decreasing? select the correct choice below and, if necessary, fill in the answer box within your choice.
○ a. the relative growth is decreasing for a >
(type an integer or a simplified fraction.)
○ b. the relative growth is decreasing for 0 < a <
(type an integer or a simplified fraction.)
○ c. the relative growth is not decreasing for any values of a.

Explanation:

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

Since \(\frac{dW}{dt}\propto W^{a}\), we can write \(\frac{dW}{dt} = kW^{a}\) (\(k>0\)).
Then \(G(W)=\frac{1}{W}\frac{dW}{dt}\). 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 \((x^{n})^\prime=nx^{n - 1}\), \(G^\prime(W)=k(a - 1)W^{a - 2}\).

• Increasing:

\(G^\prime(W)>0\). Since \(k>0\) and \(W>0\) (weight is non - negative), we need \(a-1>0\), i.e., \(a > 1\).

• Decreasing:

\(G^\prime(W)<0\). Since \(k>0\) and \(W>0\), we need \(a - 1<0\), i.e., \(0 < a<1\) (because \(a>0\) is given in the problem statement).

Answer:

For part (a), \(G(W)=kW^{a - 1}\), the relative growth rate is increasing for \(a>1\) and decreasing for \(0 < a<1\). For part (b), since the relative growth rate \(G(W)\) is decreasing as fish grow (\(W\) increases), from \(G^\prime(W)=k(a - 1)W^{a - 2}<0\) (\(k>0,W>0\)), we must have \(0 < a<1\). So for the multiple - choice question, the answer is B. The relative growth is decreasing for \(0 < a<1\).