Question

A mass of 100 grams of a particular radioactive substance decays according to the function m(t)=100e−t850, where t>0 measures time in years. When does the mass reach 30 grams?

Answer 1

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.