given the following exponential function, identify whether the change r…
This is exponential growth, with a percentage rate of increase of $21.9\%$ per unit of $x$.
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.
$y = 400e^{0.198x}$
Solution Steps
-
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.
$y = 400e^{0.198x}$ -
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: $0.2187 \times 100 = 21.87\% \approx 21.9\%$
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Final answer
This is exponential growth, with a percentage rate of increase of $21.9\%$ per unit of $x$.
Answer
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: $0.2187 \times 100 = 21.87\% \approx 21.9\%$
Answer
This is exponential growth, with a percentage rate of increase of $21.9\%$ per unit of $x$.
Question Image
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Subject
mathematics
Sub Subject
calculus
Education Level
high school
Difficulty
unspecified
Question Type
calculation
Multi Question
No
Question Count
1
Analysis Status
completed
Analyzed At
2026-02-09T20:24:28
OCR Text
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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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