Let [tex]y(t)[/tex] be the mass (in mg) remaining after [tex]t[/tex] years. Then we know the following.
[tex]
\begin{aligned}
y(t) & =y(0) e^{k t} \\
& = \\
& =\text { Enter an exact number. }
\end{aligned}
[/tex]
Answer (1)
Without additional information such as the initial mass y ( 0 ) and the rate constant k , or two points ( t 1 , y ( t 1 )) and ( t 2 , y ( t 2 )) , it is impossible to determine the exact expression for y ( t ) . The problem is incomplete and no numerical answer can be provided.
Explanation
Problem Analysis The problem states that y ( t ) is the mass (in mg) remaining after t years and provides the equation y ( t ) = y ( 0 ) e k t . However, to determine the exact expression for y ( t ) , we need more information. We need either the initial mass y ( 0 ) and the rate constant k , or two points ( t 1 , y ( t 1 )) and ( t 2 , y ( t 2 )) to solve for y ( 0 ) and k . Without additional information, we cannot proceed. Therefore, we cannot provide an exact number for y ( t ) .
Conclusion Since we lack sufficient information to determine the exact expression for y ( t ) , we cannot provide a numerical answer. The problem is incomplete.
Examples
Imagine you are tracking the decay of a radioactive substance in a lab. The equation y ( t ) = y ( 0 ) e k t models this decay, where y ( t ) is the amount of substance remaining after time t , y ( 0 ) is the initial amount, and k is the decay constant. To predict the amount of substance at any time, you need to know both the initial amount and the decay constant, highlighting the importance of complete data in real-world applications.
