Euler's Number, e Quick Check

Question

2 of 52 of 5 Items
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.

Answer 1

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