Question

1. the weight of a baby duck, $w(t)$, $t$ days after its birth can be modeled by the function $w(t)=100 - 80e^{-0.2t}$, where $w(t)$ is in grams. find the average rate of change in the baby duck’s weight over its first month of life (assume the month has 31 days). round to the nearest thousandth.

Explanation:

Step1: Recall average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b=31$, and $W(t)=100 - 80e^{-0.2t}$.

Step2: Calculate $W(31)$

Substitute $t = 31$ into $W(t)$:
$W(31)=100-80e^{-0.2\times31}=100 - 80e^{-6.2}$.
Since $e^{-6.2}\approx0.002024$, then $W(31)=100-80\times0.002024 = 100 - 0.16192=99.83808$.

Step3: Calculate $W(0)$

Substitute $t = 0$ into $W(t)$:
$W(0)=100-80e^{-0.2\times0}=100 - 80\times1=20$.

Step4: Calculate average rate of change

The average rate of change is $\frac{W(31)-W(0)}{31 - 0}=\frac{99.83808 - 20}{31}=\frac{79.83808}{31}\approx2.575$.

Answer:

$2.575$