Exponential Decay Model

Exponential Decay Problem

• Problem: An object has an initial mass of $250$ grams, and its mass halves every $5$ years. We need to find the equation that shows the number of grams, $M(t)$, remaining after $t$ years.

• Understanding Exponential Decay: Exponential decay models the decrease in quantity over time. The general form of an exponential decay equation is:
  $M(t) = M_0 (b)^{\frac{t}{T}}$
  where:

  • $M(t)$ is the mass remaining after $t$ years.
  • $M_0$ is the initial mass.
  • $b$ is the decay factor.
  • $t$ is the time in years.
  • $T$ is the time it takes for the mass to halve (half-life).

• Given Information:

  • Initial mass, $M_0 = 250$ grams.
  • The mass halves every $5$ years, so the half-life $T = 5$ years.
  • The decay factor, since the mass halves, is $b = \frac{1}{2}$.

• Setting up the Equation:
  Using the general form and the given information, the equation becomes:
  $M(t) = 250 \left(\frac{1}{2}\right)^{\frac{t}{5}}$

• Analyzing the Options:

  • (A) $M(t) = 250 (5)^{\frac{t}{4}}$
    • This is incorrect because the base is $5$ instead of $\frac{1}{2}$, and the time constant is $4$ instead of $5$.
  • (B) $M(t) = 250 (2)^{\frac{t}{5}}$
    • This is incorrect because the base is $2$ instead of $\frac{1}{2}$, indicating growth rather than decay.
  • (C) $M(t) = 250 (5)^{-2t}$
    • This is incorrect because it does not properly represent exponential decay with a half-life of $5$ years.
  • (D) $M(t) = 250 \left(\frac{1}{2}\right)^{\frac{t}{5}}$
    • This matches the derived equation and correctly represents exponential decay with an initial mass of $250$ grams halving every $5$ years.

• Conclusion: The correct equation is:
  $M(t) = 250 \left(\frac{1}{2}\right)^{\frac{t}{5}}$