Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 07, 2023

Find the rat population in 2000 given the formula $n(t)=86 e^{0.04 t}$ where $t$ is years since 2000 and $n(t)$ is in millions.

STEP 1

Assumptions
1. The rat population is given by the function $n(t)=86 e^{0.04 t}$ .
2. The variable $t$ represents the number of years since the year 2000.
3. The variable $n(t)$ represents the rat population in millions.
4. We are asked to find the rat population in the year 2000, which corresponds to $t=0$ .

STEP 2

To find the rat population in the year 2000, we need to evaluate the function $n(t)$ at $t=0$ .
$n(0)=86 e^{0.04 \times 0}$

STEP 3

Simplify the exponent since any number raised to the power of 0 is 1.
$e^{0.04 \times 0} = e^0 = 1$

STEP 4

Now, substitute the value from STEP_3 into the function to find the population.
$n(0)=86 \times 1$

STEP 5

Calculate the rat population in the year 2000.
$n(0)=86$
The rat population in the year 2000 was 86 million rats.

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