Please help! A radioactive substance decays according to the following function, where y0 is the initial amount present, and y is the amount present at time t (in days).

Question

A radioactive substance decays according to the following function, where y₀ is the initial amount present, and y is the amount present at time t (in days).

y = y₀e^-0.019t

Find the half-life of this substance. Do not round any intermediate computations, and round your answer to the nearest tenth.

Answer 1

amount at time t1

y₁=y₀e^-.019(t1)

amount at time t2

y₂=y₀e^-.019(t2)

t2>t1

want y₂/y₁=1/2

1/2=[y₀e^-.019(t2)]/[y₀e^-.019(t1)]

1/2=e^-.019(t2-t1)

t2-t1 is the half life time interval

t2-t1=(ln(1/2))/(-.019)

Answer 2

Hi Cosette,

Looking at the word problem, I first like to write down the given information

1. Radioactive Substance decay
2. y₀ is the initial amount present
3. y is the amount present at time t (in days).
4. y = y₀e^-0.019t

Next Identify the question being asked

1. Find the half-life of this substance.
2. Do not round any intermediate computations, and round your answer to the nearest tenth.

The half-life of a substance is the time it takes for half of the substance to decay. So if you started out with y₀ amount of a substance, the 1/2 life amount would be (1/2) y₀

You then can use this information in the equation to solve for t

y = y₀e^-0.019t

(1/2)y₀=y₀e^-0.019t

(1/2) = e^-0.019t (Take the ln of both sides... the ln(e)^(-0.019t) = (-0.019t)

ln(1/2) = -0.019t

ln(1/2) / (-0.019) = t

So t= 36.48143056

Rounded to the nearest 10th

t = 36.5 days

Hope this helps. Let me know if you need further clarification :)

Answer 3

y=yoe^-.019t'

half life is when

1/2 = e^-.019t

take natural logs of both sides

ln.5 = -.019t

t = ln.5/-.019

t = 36.48143 days = half life

= 36 days 11 hours, 33 minutes and about 15 1/2 seconds

Answer 4

Let's put the function of exponential decay here.

y(t) = y₀e ^-0.019t where y(t) represents the remaining amount after t time.

Half-time is the time that leaves y(t) = y_o/2. (halved)

y_o/2 = y_o e ^-0.019t_(halflife). We are solving for t_(halflife)

0.5 = e^-0.019t_(half-life) (Take ln of both sides)

ln(0.5) = -0.019t_(half-life)

t(half-life) = -ln(0.5) / 0.019 = ln(2)/0.019

t(half-life) = 36.4814305558 days ≈ 36.5 days