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1. Understand the question

   given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease per unit of $x$, to the nearest tenth of a percent.
   $y = 400e^{0.198x}$

2. Explanation

   Step1: Identify growth/decay sign

   The exponent has a positive coefficient ($0.198$), so this is exponential growth.

   Step2: Convert to percentage rate

   For continuous exponential models $y = ae^{kx}$, the growth rate $r$ is calculated as $r = e^k - 1$.
   $r = e^{0.198} - 1$
   Calculate $e^{0.198} \approx 1.2187$, so $r \approx 1.2187 - 1 = 0.2187$

   Step3: Convert to percentage

   Multiply by 100 and round to nearest tenth: $0.2187 \times 100 = 21.87\% \approx 21.9\%$

3. Final answer

   This is exponential growth, with a percentage rate of increase of $21.9\%$ per unit of $x$.

Réponse

Question Analysis

OCR Text

given the following exponential function, identify whether the change represents growth or decay, and determine the percentage rate of increase or decrease per unit of $x$, to the nearest tenth of a percent.
$y = 400e^{0.198x}$

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