Answer:
The given function is exponential decay.
Step-by-step explanation:
Given : The function below represents the number of zombies, N, where t is the number of years since the zombies gained control of Earth : [tex]N(t)=300\cdot 2^{-\frac{t}{8}}[/tex]
To find : Is this exponential growth or decay?
Solution :
Exponential function is [tex]f(x)=ab^x[/tex]
Where, a is the initial amount and b is the factor of rate.
If b>1, function has growth rate.
If b<1, function has decay rate.
We have given the function, [tex]N(t) = 300\cdot 2^{-\frac{t}{8}}[/tex]
Where, N represents the number of zombies and t is the number of years.
Applying properties of exponent,
[tex]x^{-a}=\frac{1}{x^a}[/tex]
[tex]N(t) = 300\cdot \frac{1}{2^{\frac{t}{8}}}[/tex]
[tex]N(t) = 300\cdot (\frac{1}{2})^{\frac{t}{8}}[/tex]
On comparing with general form of exponential function,
a=300 and [tex]b=\frac{1}{2}[/tex]
So, [tex]\frac{1}{2}<1\Rightarrow b<1[/tex]
Which means the function is exponential decay.
Therefore, The given function is exponential decay.
