An ancient artifact was recently discovered, but due to rust and corrosion, only 75 grams of the original item remained. Based on historical dates, scientists believe that this artifact was decaying at a rate of 2% each year. Although the artifact will now be preserved at a museum, scientists wonder: how much of the original artifact would there be if they had not discovered it for another 10 years?

Write an exponential function rule and solve. Round your answer to the nearest whole number (the ones place). Enter both the number and the associated units

[tex]\bf \qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &75\\ r=rate\to 2\%\to \frac{2}{100}\dotfill &0.02\\ t=\textit{elapsed time}\dotfill &10\\ \end{cases} \\\\\\ A=75(1-0.02)^{10}\implies A=75(0.98)^{10}\implies A\approx 61.28\implies \stackrel{\textit{rounded up}}{A=61~grams}[/tex]

JeanaShupp JeanaShupp · Answer 2 · 2019-07-26T09:24:57+02:00

Answer with explanation:

The exponential decay function is written as :-

[tex]f(x)=A(1-r)^x[/tex], where f (x) is the amount of material left after x years , A is the initial amount of material and r is the rate of decay.

Given : The amount of original item remained now = 75 grams

The rate of decay = 2% = 0.02

Now, the amount of original artifact would there be left if they had not discovered it for another 10 years is given by :-

[tex]f(10)=75(1-0.02)^{10}[/tex]

Solving the above exponential equation , we get

[tex]=61.2804605166\approx61[/tex]

Hence only 61 grams original artifact would there be left if they had not discovered it for another 10 years .