Final answer:
The decay factor 'a' for the exponential decay of titanium-51, with a half-life of 5.76 minutes, is found using the half-life formula and is approximately e^(-0.1203). This allows us to use the function A(t) = A₀ᵃʳᵗ to model the decay of titanium-51 over time.
Step-by-step explanation:
The question requires us to determine the decay factor in the exponential decay function A(t) = A₀ᵃʳ that describes the amount of a radioactive isotope, in this case, titanium-51, left after t minutes. The half-life of titanium-51 is provided as approximately 5.76 minutes. To find the decay factor a, we use the fact that after one half-life, exactly half of the original substance remains.
Therefore, after 5.76 minutes, we would have A(5.76) = A₀/2. Substituting into the exponential decay function, we have A₀/2 = A₀ᵃʳ^(5.76). Dividing both sides by A₀ and taking the natural logarithm of both sides, we can solve for a.
a = e^(ln(0.5)/5.76). After calculating the value inside the natural logarithm, we get an ≈ e^(-0.1203), which is the decay factor for titanium-51. Using this factor, we can now describe the decay of titanium-51 over time with our decay function A(t) = A₀ᵃʳᵗ, where a is approximately e^(-0.1203).