Suppose that after x billion years, the Milky Way converted...

Suppose that after $x$ billion years, the Milky Way converted half of its gas into stars. In the next $x$ billion years, it converted half of the remaining gas into stars-and so on, each $x$ billion years. The Milky Way is estimated to be about 13 billion years old. and only about $15 \%$ of its disk remains today in the form of gas.a. What value of $x$ would give the Milky Way's current gas fraction? (Hint: You can calculate this mathematically, or you can draw an approximate graph of the declining gas fraction and estimate $x$. b. In how many billion years will the gas fraction be just $1 \% ?$

Suppose that after $x$ billion years, the Milky Way converted half of its gas into stars. In the next $x$ billion years, it converted half of the remaining gas into stars-and so on, each $x$ billion years. The Milky Way is estimated to be about 13 billion years old. and only about $15 \%$ of its disk remains today in the form of
gas.a. What value of $x$ would give the Milky Way's current gas fraction? (Hint: You can calculate this mathematically, or you can draw an approximate graph of the declining gas fraction and estimate $x$.
b. In how many billion years will the gas fraction be just $1 \% ?$

Key Concepts

Exponential Decay

Exponential decay is a process in which a quantity decreases by a constant factor over equal time intervals. In modeling, it often takes the form N(t) = N?(1/2)^(t/x), where N? is the initial value and x represents the time interval over which the quantity halves. This concept is crucial for understanding processes where continuous reduction occurs, such as the conversion of gas into stars.

Half-life

Half-life refers to the time required for a quantity undergoing exponential decay to reduce to half of its initial value. It is a key concept not only in nuclear physics but also in any process where a material or substance depletes by a fixed percentage over time. In this context, the repeated halving of the gas amount in the Milky Way is analogous to a half-life process.

Logarithmic Equations

Logarithmic equations are used to solve for variables in exponential relationships. When an exponential decay model is set equal to an observed fraction, logarithms allow for isolating the time parameter. This mathematical tool is essential for calculating the number of time intervals required for a decaying substance to reach a particular fraction of its original amount.

Discrete Iterative Processes

Discrete iterative processes refer to changes that occur at specific intervals rather than continuously. This concept is important when the updating or transformation event happens in distinct steps, as in the problem where the gas amount is halved every x billion years. Modeling such processes requires understanding how each iterative step affects the overall system over time.

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