A radioactive substance decays from 85mg to 22.1mg in 25 years according to the exponential decay model y=ae-bx, where a is the initial amount and y is the amount remaining after years.

Question

A radioactive substance decays from 85mg to 22.1mg in 25 years according to the exponential decay model y=ae^-bx, where a is the initial amount and y is the amount remaining after years.

 

Find the b-value. 

Round to the nearest hundredth, if needed.

b=   ?  

Use the EXACT b-value to write the exponential decay model for this substance with initial amount 85 mg, then use that model to find the half-life.

Find the half-life.

Round to the nearest hundredth of a year, if needed.

half-life ≈ __?__years

Answer 1

Given y = a• e^{–b • x}

(22.1 / 85) = e^–25•b

85/22.1 = e^25•b

3.84615 ⁼ e^25•b

ln 3.84615 = 25•b

b = 1/25 (1.347) = 0.05388

T_1/2 = 0.693/ b = 12.86 years

Answer 2

Substitute the given values in the exponential decay model y = a• e^{–b • x}

22.1 = 85• e^{–b• 25}

(22.1 / 85) = e^–25•b

^{Take natural log on both sides to get the exact value of b =[ ln(22.1/85)] /(–25)}

^{Use your calculator and round it to the hundredth place value.}

^{Half life Using the same exponential decay model and substitute}

^{(85/2) = 85 e^(–exact value of b)•x Simplifying this,}

^{(1/2) = e ^(-exact value of b)•x}

^{Take natural log on both sides and simplify.}

^{Round the value of x to the nearest hundredth. (educated guess - this value will be about one half of the value of b)}