Question

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The function f(t): 8000(0.2) represents the change in a quantity over t years.
What does the constant 0.2 reveal about the rate of change of the quantity?

Answer 1

Answer and Explanation:

We are given the exponential decay function:

$f(t) = 8000(0.2)^t$

If we examine how this function changes as t increases:

$\begin{array}{| c | c |}\cline{1-2} t & f(t) \\ \cline{1-2} 0 & 8000 \\ \cline{1-2} 1 & 1600 \\ \cline{1-2} 2 & 320 \\ \cline{1-2} 3 & 64\\ \cline{1-2} \end{array}$

We can see that:

• for every t increase of 1,
• the function becomes 0.2 of its current value.

For example, from t = 1 to t = 2,

f(t) goes from:

$f(1) = 8000(0.2)^1 = 1600$

to

$f(2) = 8000(0.2)^2\\ \\ \text{ }\ \ \ \ \: = \left[\!\dfrac{}{}8000(0.2)\dfrac{}{}\!\right]\!(0.2) \\ \\ \text{ }\ \ \ \ \: = 1600(0.2) \ \ \text{ or }\ \ \left[\!\dfrac{}{}f(1)\dfrac{}{}\!\right]\!(0.2) \\ \\ \text{ }\ \ \ \ \: = 320$

So, the constant 0.2 shows that the rate of the function's decay is proportional to its current size by a factor of 0.2.