Question

certain radioactive material decays in such a way that the mass remaining after t years is given by the function m(t)=455e^{-0.045t} where m(t) is measured in grams. (a) find the mass at time t = 0. your answer is (b) how much of the mass remains after 50 years? your answer is round answers to 1 decimal place.

Explanation:

Step1: Substitute \(t = 0\) into the function

Given \(m(t)=455e^{- 0.045t}\), when \(t = 0\), we have \(m(0)=455e^{-0.045\times0}\).
Since \(a^0 = 1\) for \(a
eq0\), here \(e^{0}=1\). So \(m(0)=455\times1 = 455\).

Step2: Substitute \(t = 50\) into the function

When \(t = 50\), \(m(50)=455e^{-0.045\times50}\).
First, calculate \(-0.045\times50=-2.25\). Then \(m(50)=455e^{-2.25}\).
We know that \(e^{-x}=\frac{1}{e^{x}}\), and \(e^{2.25}\approx9.4877\). So \(m(50)=455\times\frac{1}{9.4877}\approx48.0\).

Answer:

(a) \(455\)
(b) \(48.0\)