Question

The half-life of a radioactive substance is the amount of time required for half its mass to decay. Suppose we have 41 grams of the radioactive substance Krypton. Assume that it decays continuously at an annual rate of 6.5%.

Write a function
f
that determines Krypton's mass (in grams) in terms of the number of years
t
since we measured its initial amount.

Answer 1

Answer:

$N(t)=41\, \,e^{-0.065*t}$

Explanation:

We use exponential decay for this example, considering that the initial amount of substance is 41 grams, and the continuous rate of decay is 6.5%:

$N(t)=N_0\, \,e^{-k*t}\\N(t)=41\, \,e^{-0.065*t}$

where N(t) is the Krypton's mass of the sample in grams, in terms of number of years "t"

Answer 2

Using the exponential distribution concept, the expression which models the function can be expressed thus : $N(t) = 41 e^{0.065t}$

Given the Parameters :

• Initial mass of Krypton = 41
• Decay rate, r = 6.5% = 0.065

Using the exponential decay relation :

• $N(t) = N_{0} e^{rt}$
• $final \: amount = N(t)$
• $Initial \: mass = N_{0}$
• $time = t$
• $Decay \: rate = r$

Hence, following the exponential Decay relation ; the function which models the mass of krypton based on the initial amount, decay rate and time can be expressed thus :

• $N(t) = 41 e^{0.065t}$

Therefore, the function which models the scenario is $N(t) = 41 e^{0.065t}$

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