Euler's Number, e Quick Check

Question

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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.

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