Westonci.ca is the premier destination for reliable answers to your questions, brought to you by a community of experts. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
We are given that a radio isotope has an initial mass of 60 grams and a half-life of 11 years. The mass of the isotope with respect to time can be modeled using the following exponential function:
[tex]M=m_0e^{kt}[/tex]Where:
[tex]\begin{gathered} M=\text{ mass} \\ m_0=\text{ initial mass} \\ k=\text{ constant} \\ t=\text{ time} \end{gathered}[/tex]Now, since we are given that the half-life is 11 years this means that the mass will be half of the initial mass when "t = 11 years". We can use this fact to calculate the value of "k". Substituting half the initial mass we get:
[tex]\frac{m_0}{2}=m_0e^{k(11\text{year)}}[/tex]Now, we may cancel the initial mass:
[tex]\frac{1}{2}=e^{k(11\text{years)}}[/tex]Now, we solve for "k". To do that we will take natural logarithm to both sides:
[tex]\ln \frac{1}{2}=\ln e^{k(11\text{years)}}[/tex]Now, we use the following property of logarithms:
[tex]\ln x^y=y\ln x[/tex]Applying the property we get:
[tex]\ln \frac{1}{2}=k(11years)\ln e^{}[/tex]We have that:
[tex]\ln e=1[/tex]Therefore:
[tex]\ln \frac{1}{2}=k(11years)[/tex]Now, we divide both sides by 11:
[tex]\frac{1}{11}\ln \frac{1}{2}=k[/tex]Solving the operations:
[tex]-0.063=k[/tex]Now, we substitute in the formula for the mass:
[tex]M=60e^{-0.063t}[/tex]Now, to determine the value of the mass after 25 years we substitute the value "t = 25", we get:
[tex]M=60e^{-0.063(25)}[/tex]Solving the operations:
[tex]M=12.4[/tex]Therefore, the mass after 25 years is 12.4 grams.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
