Question

The function below represents the number of zombies, N, where t is the number of years since the zombies gained control of Earth: N(t) = 300 · 2-t/8 Is this exponential growth or decay? Explain using your understanding of the properties of exponents. (To type exponents, use the ^ key.)

Answer 1

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 : $N(t)=300\cdot 2^{-\frac{t}{8}}$

To find : Is this exponential growth or decay?

Solution :

Exponential function is $f(x)=ab^x$

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,  $N(t) = 300\cdot 2^{-\frac{t}{8}}$

Where, N represents  the number of zombies and t is the number of years.

Applying properties of exponent,

$x^{-a}=\frac{1}{x^a}$

$N(t) = 300\cdot \frac{1}{2^{\frac{t}{8}}}$

$N(t) = 300\cdot (\frac{1}{2})^{\frac{t}{8}}$

On comparing with general form of exponential function,

a=300 and $b=\frac{1}{2}$

So, $\frac{1}{2}$

Which means the function is exponential decay.

Therefore, The given function is exponential decay.

Answer 2

This is decay. The exponent has a negative sign. The reciprocal of 2 is 0.5. So N(t) can be rewritten as 0.5^(t/8). The base is fraction less than 1, which is decay.