Final answer:
To find when the mass of the radioactive substance reaches 30 grams, we solve the equation 30 = 100e^(-t/850) for t using algebra and the properties of logarithms. This involves dividing by 100, taking the natural logarithm of both sides, and then solving for t.
Step-by-step explanation:
To determine when the mass of a radioactive substance reaches 30 grams given the decay function m(t)=100e−t/850, where t is the time in years, we need to solve for t when m(t) equals 30 grams.
We set the decay function equal to 30:
30 = 100e−t/850
Dividing both sides by 100 gives us:
0.3 = e−t/850
To solve for t, we take the natural logarithm of both sides:
ln(0.3) = −t/850
Multiplying both sides by −850 to isolate t yields:
t = −850 × ln(0.3)
Calculating this gives us the value for t, which is the time it takes for the mass to reach 30 grams.