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

Answer:

It will take approximately 1657.3 years.

Step-by-step 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,

$\Rightarrow 20=24\cdot e^{-0.00011t}$

$\Rightarrow e^{-0.00011t}=\dfrac{20}{24}$

$\Rightarrow \ln e^{-0.00011t}=\ln \dfrac{20}{24}$

$\Rightarrow {-0.00011t}\times \ln e=\ln \dfrac{20}{24}$

$\Rightarrow {-0.00011t}\times 1=\ln \dfrac{20}{24}$

$\Rightarrow {-0.00011t}=\ln \dfrac{20}{24}$

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

$\Rightarrow 0.00011t=0.1823$

$\Rightarrow t=\dfrac{0.1823}{0.00011}=1657.3\ years$