Final answer:
To find the half-life of the radioactive material with the equation A(t) = 116e^-0.44t, we solve the equation 0.5 = e^(-0.44t1/2) for t1/2, which yields a half-life of approximately 1.57 hours.
Step-by-step explanation:
To find the half-life of a radioactive material described by the equation A(t) = 116e-0.44t, we need to determine the time it takes for the radioactivity to decrease to half its original value. The half-life (t1/2) is the time for the activity to reach A(t)/2. Set up the equation 58 = 116e-0.44t1/2 and solve for t1/2, since 58 is half of 116.
Divide both sides by 116 to get 0.5 = e-0.44t1/2, take the natural logarithm of both sides to get ln(0.5) = -0.44t1/2, which gives us t1/2 = ln(0.5) / -0.44. Calculating this using a calculator or logarithm tables, we find the half-life is approximately 1.57 hours.