Question

Hal is asked to write an exponential function to represent the value of a $10,000 investment decreasing at 2% annually. What multiplicative rate of change should Hal use in his function?

Answer 1

Let P(t) be the amount of investment over time t. We are given that the investment IS decreasing at 2% annual rate. That means after the first year, the decrease in $10000 will be 10000(-0.02) = $200. So the amount left after the first year is 10000 - 200 = 9800.

For exponential growth or decrease there is a general formula given by P(t) = P(0) [exp(rt)]
where,
P(t) is the investment remaining at time t,
P(0) is the investment at time 0, initial investment and
r is the exponential rate of decrease.

We have, with t = 1 (after 1 year), P(1) = 9800, P(O) = 10000. Substituting in the formula, we get

9800 = 10000[exp(r)], Note r times 1 is r.

That is 0.98 = exp(r),
so
r = loge (0.98) = ln (0.98) = -0.0202, to five decimals. Hence the formula for the exponential function Hal is asked to write is

P(t) = P(0)[exp(-0.0202t)].

Answer 2

would the multiplicative for the function be -0.0202

Answer 3

\(f(t) = .98^{t}\) or the exponential form that has been given.