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. for which values of a is the relative growth rate increasing? select the correct choice below and, if necessary, fill in the answer box within your choice. a. the relative growth is increasing for $ 0

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

Accepted Answer

# Explanation: ## Step1: Express $G(W)$ Given $\frac{dW}{dt}\propto W^{a}$, we can write $\frac{dW}{dt} = kW^{a}$ ($k>0$). Then $G(W)=\frac{1}{W}\frac{dW}{dt}$. Substituting $\frac{dW}{dt} = kW^{a}$ into the formula for $G(W)$, we get $G(W)=\frac{kW^{a}}{W}=kW^{a - 1}$. ## Step2: Find the derivative of $G(W)$ Differentiate $G(W)$ with respect to $W$ using the power rule $(x^{n})^\prime=nx^{n - 1}$. So $G^\prime(W)=k(a - 1)W^{a - 2}$. ## Step3: Determine when $G(W)$ is increasing A function $y = G(W)$ is increasing when $G^\prime(W)>0$. Since $k>0$ and $W>0$ (weight is non - negative and we consider $W>0$ for growth analysis), we need to consider the sign of $(a - 1)$. - If $a-1>0$ (i.e., $a > 1$), then $G^\prime(W)>0$ for $W>0$. - If $a - 1<0$ (i.e., $a<1$), then $G^\prime(W)<0$ for $W>0$. # Answer: B. The relative growth is increasing for $a>1$

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Question

Explanation:

Step1: Express \(G(W)\)

Step2: Find the derivative of \(G(W)\)

Step3: Determine when \(G(W)\) is increasing

Answer: