Question

Plutonium-240 decays according to the function where Q represents the quantity remaining after t years and k is the decay constant, 0.00011... How long will it take 24 grams of plutonium-240 to decay to 20 grams?

Answer 1

Note: a radioactive decay constant is always negative.

time = [natural log(ending amount / beginning amount)] / k
time = ln (20 / 24) / -.00011
time = ln (5/6) / -.00011
time = -.018232155683 / -.00011
time = 165.7468698455
time = 165.75 years

Answer 2

It will take approximately 1657.3 years.

Explanation:

The function that defines the exponential decay for this system is,

`Q(t)=Q_0\cdot e^(-kt)`

Q(t) = The amount after time t = 20

Q₀ = Initial amount = 24

k = Decay constant = 0.00011

t = time

Putting the values,

`\Rightarrow 20=24\cdot e^(-0.00011t)`

`\Rightarrow e^(-0.00011t)=(20)/(24)`

`\Rightarrow \ln e^(-0.00011t)=\ln (20)/(24)`

`\Rightarrow {-0.00011t}* \ln e=\ln (20)/(24)`

`\Rightarrow {-0.00011t}* 1=\ln (20)/(24)`

`\Rightarrow {-0.00011t}=\ln (20)/(24)`

`\Rightarrow {-0.00011t}=-0.1823`

`\Rightarrow 0.00011t=0.1823`

`\Rightarrow t=(0.1823)/(0.00011)=1657.3\ years`