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6. Determine the initial amount of radioactive substance if after 22 years, the remaining unconverted substance is 50mg and after another 10 years the remaining unconverted substance is 40mg. Determine the decay rate factor.

6. Determine the initial amount of radioactive substance if after 22 years, the remaining unconverted substance is 50mg and after another 10 years the remaining unconverted substance is 40mg. Determine the decay rate factor.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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6. Determine the initial amount of radioactive substance if after 22 years, the remaining unconverted substance is 50mg and after another 10 years the remaining unconverted substance is 40mg. Determine the decay rate factor.
Transcribed Image Text:6. Determine the initial amount of radioactive substance if after 22 years, the remaining unconverted substance is 50mg and after another 10 years the remaining unconverted substance is 40mg. Determine the decay rate factor.
NATURAL GROWTH AND DECAY These problems involve time rate of change of variable that is proportional to the same variable being referred to. If x represents this variable, then the differential equation is dx = kx dt Solving this equation dx k dt In x = kt + c x = ekt+c x = cekt At initial condition, time t = 0, x = xo Xo = cek(0) c = Xo Then the general solution or formula for this type of problems is x = x,ekt The following summarizes the formulas and their variables for each problem Natural Population Growth Continuous Compounding Interest P = Poekt A = Aoet P – population at any time, t A – amount at any time, t Po – population at t = 0, initial condition Ao – amount deposited at t = 0 k – growth rate factor, k > 0 i- annual rate of continuously compounded interest Radioactive Decay Half-Life of a Radioactive Material x = x,e-kt x = x,e-kt x – amount of unconverted substance present Att = 1 (half-life), x =÷xo 0, initial condition In 2 T=- k Xo - amount present at t = k – decay constant, k > 0
Transcribed Image Text:NATURAL GROWTH AND DECAY These problems involve time rate of change of variable that is proportional to the same variable being referred to. If x represents this variable, then the differential equation is dx = kx dt Solving this equation dx k dt In x = kt + c x = ekt+c x = cekt At initial condition, time t = 0, x = xo Xo = cek(0) c = Xo Then the general solution or formula for this type of problems is x = x,ekt The following summarizes the formulas and their variables for each problem Natural Population Growth Continuous Compounding Interest P = Poekt A = Aoet P – population at any time, t A – amount at any time, t Po – population at t = 0, initial condition Ao – amount deposited at t = 0 k – growth rate factor, k > 0 i- annual rate of continuously compounded interest Radioactive Decay Half-Life of a Radioactive Material x = x,e-kt x = x,e-kt x – amount of unconverted substance present Att = 1 (half-life), x =÷xo 0, initial condition In 2 T=- k Xo - amount present at t = k – decay constant, k > 0
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