Question

A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

Answer 1

Answer:

Part 1 : $A=250 e^{-0.008223t}$

Part 2 : Half life is 84 minutes ( approx )

Step-by-step explanation:

Part 1 : Suppose the function that shows the amount( in grams ) of the substance after t minutes,

$A=A_0 e^{kt}$

If t = 0 min, A = 250 grams,

$250=A_0 e^{0}$

$\implies A_0 = 250$

If t = 250, A = 32 grams,

$32 = A_0 e^{250k}$

$32 = 250 e^{250k}$

$0.128 = e^{250k}$

Taking ln both sides,

$\ln(0.128) = 250k$

$\implies k =\frac{\ln(0.128)}{250}=-0.008223$

Hence, the equation that shows this situation,

$A=250 e^{-0.008223t}$

Part 2 : If A = 250/2 = 125,

$125 = 250 e^{-0.008223t}$

$0.5 = e^{-0.008223t}$

Taking ln both sides,

$\ln(0.5) = -0.008223t$

$\implies t =\frac{\ln(0.5)}{-0.008223}\approx 84$

Therefore, the half life of the substance would be 84 minutes.