# This is a multiple choice question. A certain radioactive substance is decaying so that at time t, measured in years, the amount of the substance, in grams, is given by the function f(t)=-e^t +10. what is the rate of decay of the substance after 1 year?

## the choice are a) -3 g/year b) -ln 2 g/year c) -2.72 g/year d)-1.53 g/year

Apr 21, 2018

C.

#### Explanation:

If the amount of substance left is given by $f \left(t\right) = - {e}^{t} + 10 ,$ the rate of decay will be given by the first derivative of that function.

So, find the first derivative, recalling that the derivative of the natural exponential function ${e}^{t}$ is just itself, and that the $10 ,$ a constant, goes away, as the derivative of a constant is zero.

$f ' \left(t\right) = - {e}^{t}$ (Units: g/year)

To determine the rate of decay after $1$ year, evaluate the derivative at $t = 1 :$

$f ' \left(1\right) = - {e}^{1} \approx - 2.72$ g/year