Questions: The weight of a baby duck, W(t), t days after its birth can be modeled by the function W(t)=100-80 e^-0.2 t. 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. W(t)=100-80 e^-0.2 t w(31)=100-80 e^-0.2(31)

The weight of a baby duck, W(t), t days after its birth can be modeled by the function W(t)=100-80 e^-0.2 t. 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. W(t)=100-80 e^-0.2 t w(31)=100-80 e^-0.2(31)
Transcript text: The weight of a baby duck, $W(t), t$ days after its birth can be modeled by the function $W(t)=100-80 e^{-0.2 t}$. 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. \[ \begin{array}{l} W(t)=100-80 e^{-0.2 t} \\ w(31)=100-80 e^{-0.2(31)} \end{array} \]
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Solution

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Solution Steps

To find the average rate of change in the baby duck's weight over its first month of life, we need to calculate the change in weight from day 0 to day 31 and then divide by the number of days (31). The average rate of change is given by the formula:

\[ \text{Average Rate of Change} = \frac{W(31) - W(0)}{31 - 0} \]

We will first compute \(W(0)\) and \(W(31)\) using the given function \(W(t) = 100 - 80e^{-0.2t}\), and then use the formula above to find the average rate of change.

Step 1: Define the Weight Function

The weight of the baby duck \( W(t) \) days after its birth is given by the function: \[ W(t) = 100 - 80e^{-0.2t} \]

Step 2: Calculate \( W(0) \)

To find the weight at birth (\( t = 0 \)): \[ W(0) = 100 - 80e^{-0.2 \cdot 0} = 100 - 80 \cdot 1 = 20.0 \, \text{grams} \]

Step 3: Calculate \( W(31) \)

To find the weight after 31 days (\( t = 31 \)): \[ W(31) = 100 - 80e^{-0.2 \cdot 31} \approx 100 - 80 \cdot e^{-6.2} \approx 99.8376 \, \text{grams} \]

Step 4: Calculate the Average Rate of Change

The average rate of change in the baby duck's weight over the first 31 days is given by: \[ \text{Average Rate of Change} = \frac{W(31) - W(0)}{31 - 0} = \frac{99.8376 - 20.0}{31} \approx 2.575 \, \text{grams per day} \]

Final Answer

\[ \boxed{2.575 \, \text{grams per day}} \]

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