A sample of radioactive material decays over time (measured in hours) with decay

Question

A sample of radioactive material decays over time (measured in hours) with decay constant.5. The graph of the exponential function y P(t), shown on the right, gives the number of grams remaining after t hours (a)How much was remaining after 1 hour? (b)Approximate the half-life of the material (c)How fast was the sample decaying after 5 hours? (d)When was the sample decaying at the rate of .2 grams per hour? (a) After 1 hour there were g remaining Round to the nearest integer as needed.) (b) The half-life was at approximately hrs. (Round to one decimal place as needed.) grams per hour (c) The sample was decaying at a rate of Round to one decimal place as needed.) (d) The sample was decaying at the rate of .2 grams per hour afterhrs Round to the nearest integer as needed.)

Explanation / Answer

equation of the graph is y=P(t) =8e-0.5t

(a)

after 1 hour,

amount remaining =8e-0.5*1

=>amount remaining = 4.852245277701067

After 1 hour there were 5 g remaining

(b)

at first half-life ,amount remaining =8/2 =4 g

=>8e-0.5t=4

=>e0.5t=2

=>0.5t=ln(2)

=>t=2ln(2)

=>t=1.3862943611198906188344642429164

The half-life at approximately 1.4 hours

(c)

y=P(t) =8e-0.5t

=>dy/dt=P'(t) =8(-0.5)e-0.5t

=>dy/dt=P'(t) =-4e-0.5t

after 5 hours ,

P'(5) =-4e-0.5*5

=>P'(5) = -0.3283399944956

The sample was decaying at a rate of 0.3 grams per hour

(d)

P'(t)=-0.2

=>-4e-0.5t=-0.2

=>e0.5t=-4/-0.2

=>e0.5t=20

=>0.5t=ln(20)

=>t=2ln(20)

=>t=5.9914645471079819868704471522851

The sample was decaying at the rate of 0.2 grams per hour after 6 hours

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