The decay formula is \( M(t) = M_0 \left(\frac{1}{2}\right)^{t/T_{1/2}} \), where \( M_0 = 50 \), \( T_{1/2} = 10 \), and \( t = 25 \).

["### The Decay Formula Explained: How Half-Life Applies to Radioactive Decay (M(t) = M₀ × (½)^(t/T₁/₂))", "Understanding radioactive decay is essential in physics, chemistry, and environmental science. One of the most fundamental equations used to describe exponential decay is:", "[\nM(t) = M_0 \left(\frac{1}{2}\right)^{t / T_{1/2}}\n]", "In this formula, M(t) represents the remaining mass (or quantity) of a radioactive substance at time ( t ), M₀ is the initial mass, T₁/₂ is the half-life — the time it takes for half the substance to decay — and ( t ) is the elapsed time.", "---", "### What Does This Formula Mean?", "The decay of radioactive materials follows an exponential law: with each half-life, the quantity of the substance reduces by half. This predictable pattern allows scientists to estimate how much material remains after any given period, making it invaluable in fields like nuclear medicine, archaeology (via carbon dating), and energy production.", "---", "### Plugging in Real Values", "Given:\n- Initial mass, ( M_0 = 50 )\n- Half-life, ( T_{1/2} = 10 )\n- Elapsed time, ( t = 25 )", "We substitute into the formula:", "[\nM(25) = 50 \left(\frac{1}{2}\right)^{25 / 10}\n]", "Simplify the exponent:", "[\nM(25) = 50 \left(\frac{1}{2}\right)^{2.5}\n]", "Now calculate the fraction:", "[\n\left(\frac{1}{2}\right)^{2.5} = 2^{-2.5} \approx 0.17678\n]", "Multiply by initial mass:", "[\nM(25) = 50 \ imes 0.17678 \approx 8.839\n]", "---", "### Result and Interpretation", "After 25 half-lives (each 10 years, so 250 years total), approximately 8.84 units of the original 50 units remain. Although this number seems small, it illustrates how fast radioactive decay proceeds — especially over longer timescales.", "---", "### Why This Formula Matters", "- Predictability: Helps plan storage and disposal for nuclear waste.\n- Accuracy: Provides reliable quantitative estimates beyond simple visual inspection.\n- Versatility: Applies beyond radioactivity — useful in finance (compound decay models), population studies, and more.", "---", "### Final Thoughts", "The decay formula ( M(t) = M_0 \left(\frac{1}{2}\right)^{t / T_{1/2}} ) is a cornerstone of understanding decay processes. With clear initial values like ( M_0 = 50 ) and ( T_{1/2} = 10 ), even intermediate times like ( t = 25 ) can be computed precisely — yielding insights critical to scientific and industrial applications.", "If you're modeling dynamic processes involving decay, remember: smaller ( t ) means higher remaining material, while longer ( t ) brings quantities close to zero. Use this knowledge confidently — and calculate with precision!", "---", "Keywords: radioactive decay formula, half-life decay, exponential decay calculation, M(t) formula, M₀ initial mass, T₁/₂ half-life, decay chemistry, nuclear physics, practical decay equations, exponential decay example", "---", "Note: For exact accuracy, consider using logarithmic functions or a calculator when dealing with non-integer exponents — but basic hand calculations with fractional powers like above offer valuable insight into decay dynamics."]