Question
Euler's Number, e Quick Check
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Question
The percentage of area damaged in a fire after t hours of burning is modeled by P(t)=12e0.08t
. To the nearest whole number, what percentage of area will be damaged by a fire that burns for eight hours?(1 point)
Responses
21 percent
21 percent
23 percent
23 percent
13 percent
13 percent
This is impossible to determine with the information provided.
Answers
GPT-4o mini
To calculate the percentage of area damaged by the fire after \( t = 8 \) hours using the model \( P(t) = 12e^{0.08t} \), we can plug in \( t = 8 \):
\[
P(8) = 12e^{0.08 \times 8}
\]
First, calculate \( 0.08 \times 8 \):
\[
0.08 \times 8 = 0.64
\]
Now, substitute this back into the equation:
\[
P(8) = 12e^{0.64}
\]
Next, calculate \( e^{0.64} \):
Using a calculator, \( e^{0.64} \approx 1.8573 \).
Now substitute this value back into the equation:
\[
P(8) = 12 \times 1.8573 \approx 22.2876
\]
To find the nearest whole number:
\[
P(8) \approx 22
\]
So the area damaged by the fire after 8 hours is approximately 22 percent.
Among the given responses, the closest option is:
**21 percent**
\[
P(8) = 12e^{0.08 \times 8}
\]
First, calculate \( 0.08 \times 8 \):
\[
0.08 \times 8 = 0.64
\]
Now, substitute this back into the equation:
\[
P(8) = 12e^{0.64}
\]
Next, calculate \( e^{0.64} \):
Using a calculator, \( e^{0.64} \approx 1.8573 \).
Now substitute this value back into the equation:
\[
P(8) = 12 \times 1.8573 \approx 22.2876
\]
To find the nearest whole number:
\[
P(8) \approx 22
\]
So the area damaged by the fire after 8 hours is approximately 22 percent.
Among the given responses, the closest option is:
**21 percent**