Questions: A new book goes on sale, and its sales t weeks after being released are modeled by f(t)=26.1+0.85 ln (t+3), where f(t) is measured in millions of books. What was the rate of change in sales after 8 weeks? (Round your answer to three decimal places.)

A new book goes on sale, and its sales t weeks after being released are modeled by f(t)=26.1+0.85 ln (t+3), where f(t) is measured in millions of books. What was the rate of change in sales after 8 weeks? (Round your answer to three decimal places.)

Transcript text: A new book goes on sale, and is sales $t$ weeks after being released are modeled by $f(t)=26.1+0.85 \ln (t+3)$, where $f(t)$ is measured in millions of books. What was the rate of change in sales after 8 weeks? (Round your answer to three decimal places.)

Solution

Solution Steps

To find the rate of change in sales after 8 weeks, we need to calculate the derivative of the sales function $ f(t) $ and then evaluate it at $ t = 8 $. The derivative will give us the rate of change of sales with respect to time.

Step 1: Define the Sales Function

The sales of the book $ t $ weeks after being released are modeled by the function: \[ f(t) = 26.1 + 0.85 \ln(t + 3) \]

Step 2: Calculate the Derivative

To find the rate of change in sales, we need to calculate the derivative of $ f(t) $ with respect to $ t $: \[ f'(t) = \frac{0.85}{t + 3} \]

Step 3: Evaluate the Derivative at $ t = 8 $

Substitute $ t = 8 $ into the derivative to find the rate of change in sales after 8 weeks: \[ f'(8) = \frac{0.85}{8 + 3} = \frac{0.85}{11} \approx 0.07727 \]

Final Answer

$\boxed{0.077}$

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