Asked by Sara
The amount A in grams of radioactive plutonium remaining in a 20-gram sample after t days is given by the formula A=20*(1/2)^(t/140). At what rate is the plutonium decaying when t=2 days?
Answers
Answered by
MathMate
Given
A(t)=20(1/2)^(t/140)
We look for A'(t)
Use implicit differentiation.
Take ln on both sides
ln(A)=ln 20 + (t/140)ln(1/2)
differentiate with respect to t
(1/A)dA/dt = 0 + ln(1/2)/140
substitute A
dA/dt = ln(1/2)/140*20(1/2)^(t/140)
substitute -ln(2) for ln(1/2)
dA/dt = -(ln(2)/7)*(1/2)^(t/140)
A(t)=20(1/2)^(t/140)
We look for A'(t)
Use implicit differentiation.
Take ln on both sides
ln(A)=ln 20 + (t/140)ln(1/2)
differentiate with respect to t
(1/A)dA/dt = 0 + ln(1/2)/140
substitute A
dA/dt = ln(1/2)/140*20(1/2)^(t/140)
substitute -ln(2) for ln(1/2)
dA/dt = -(ln(2)/7)*(1/2)^(t/140)
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