the amount of water a new tree needs to survive is related to its size. suppose a tree requires (w(g)=30 +…

Explanation:

Step1: Apply chain - rule

The chain - rule states that $\frac{dw}{dt}=\frac{dw}{dg}\cdot\frac{dg}{dt}$. First, find $\frac{dw}{dg}$ and $\frac{dg}{dt}$.

Step2: Differentiate $w(g)$ with respect to $g$

Given $w(g)=30 + 13\ln(g)$. Using the derivative formula $\frac{d}{dg}\ln(g)=\frac{1}{g}$ and $\frac{d}{dg}(c)=0$ (where $c$ is a constant), we have $\frac{dw}{dg}=\frac{13}{g}$.

Step3: Differentiate $g(t)$ with respect to $t$

Given $g(t)=2t$. Using the power - rule $\frac{d}{dt}(at)=a$ (where $a$ is a constant), we get $\frac{dg}{dt}=2$.

Step4: Substitute $g(t)$ into $\frac{dw}{dg}$

Since $g = 2t$, then $\frac{dw}{dg}=\frac{13}{2t}$.

Step5: Calculate $\frac{dw}{dt}$

By the chain - rule $\frac{dw}{dt}=\frac{dw}{dg}\cdot\frac{dg}{dt}=\frac{13}{2t}\cdot2=\frac{13}{t}$.

Step6: Evaluate at $t = 3$

Substitute $t = 3$ into $\frac{dw}{dt}$. We have $\frac{dw}{dt}\big|_{t = 3}=\frac{13}{3}\approx4.33$.

Answer:

4.33