At t=0, the mass of the radioactive material is 500 grams. After 20 years, 301.9 grams remain. The bacteria population exceeds 3639 after approximately 10.2 years.
To answer the student's questions regarding the exponential decay of a radioactive material and the exponential growth of a bacteria population, we use the given formulas and substitute the respective values.
Radioactive Material Decay
(a) To find the mass of the radioactive material at time t=0, we substitute 0 for t in the equation m(t) = 500e^-0.025t.
This simplifies to m(0) = 500 grams since e^0 = 1.
(b) To find the mass remaining after 20 years, plug in t = 20 into the equation: m(20) = 500e^(-0.025*20).
Compute this to get 301.9 grams (rounded to one decimal place).
Bacteria Population Growth
To estimate when the population will exceed 3639, we solve the inequality P(t) > 3639.
This can be rearranged to find the value of t where P(t) = 1750e^0.12t surpasses 3639.
The equation simplifies to e^0.12t > 3639/1750, which after taking logarithms gives us t > (ln (3639/1750)) / 0.12.
This results in t > 10.2 years (rounded to one decimal place).
The probable question may be:
Certain radioactive material decays in such a way that the mass remaining after t years is given by the function
m(t) = 500e-0.025t
where m(t) is measured in grams.
(a) Find the mass at time t=0
Your answer is________
(b) How much of the mass remains after 20 years?
Your answer is_________
Round answers to 1 decimal place.
A population of bacteria is growing according to the equation P(t) = 1750e^{0.12t}. Estimate when the population will exceed 3639.
t =____________
Give your answer accurate to at least one decimal place.