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}}