Final answer:
The decay of a radioactive material can be modeled using either the function m(t) = m0(2^-t/h) or the function m(t) = m0e^-rt. To find the mass remaining after a certain number of years, substitute the given value of t into the equation. The mass remaining after 82 years can be found by substituting t = 82 into m(t) = m0(2^-t/h). To find the number of years when only 2 g of the sample remain, set m(t) = 2 in m(t) = m0e^-rt and solve for t.
Step-by-step explanation:
(a) Function m(t) = m0(2-t/h) that models the mass remaining after t years:
The decay of a radioactive material can be modeled using the formula m(t) = m0(2-t/h), where m(t) is the mass remaining after t years, m0 is the initial mass, and h is the half-life of the material.
(b) Function m(t) = m0e-rt that models the mass remaining after t years:
Another way to model the decay of a radioactive material is using the formula m(t) = m0e-rt, where m0 is the initial mass, r is the decay constant, and t is the time in years.
(c) How much of the sample will remain after 82 years:
To find the mass remaining after 82 years, we can substitute t = 82 into the equation m(t) = m0(2-t/h). Calculate m(82) = 13(2-82/30) to find the remaining mass, rounded to one decimal place.
(d) After how many years will only 2 g of the sample remain:
To find the number of years when only 2 g of the sample will remain, we can use the equation m(t) = m0e-rt and set m(t) = 2. Solve for t using the formula t = (1/r)ln(m0/m).