The decay factor of an exponential function $P ( t )$ is 0.68. The initial value is 1.45. Then a formula for P is
A) $\rho = 0.68 \times t ^ { 1.45 }$
B) $P = 0.68 \times 1.45 ^ {t }$
C) $P = 1.45 \times t ^ { 0.68 }$
D) $P = 1.45 \times 0.68 ^ { t }$

Question

The decay factor of an exponential function $P ( t )$ is 0.68. The initial value is 1.45. Then a formula for P is 
A) $\rho = 0.68 \times t ^ { 1.45 }$
B) $P = 0.68 \times 1.45 ^ {t }$
C) $P = 1.45 \times t ^ { 0.68 }$
D) $P = 1.45 \times 0.68 ^ { t }$

Quizplus · Accepted Answer

The decay factor of an exponential function is given by the formula: $\rho = e ^ { -kt }$, where $\rho$ is the decay factor, $k$ is the decay constant, and $t$ is the time. Solving for $k$ we get $k = -\frac{\ln \rho}{t}$. Plugging in $\rho = 0.68$, we get $k = -\frac{\ln 0.68}{t}$. The formula for $P$ is $P = ae ^ {kt}$, where $a$ is the initial value. Plugging in $a = 1.45$ and $k = -\frac{\ln 0.68}{t}$, we get $P = 1.45 \times 0.68 ^ t$. Therefore, the correct answer is D.

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