Help Please! A radioactive substance decaying so that after t years, the amount remaining, expressed as a percent of the original amount is a(t) = 100(1.1)^-t. determine the function, a', which represents the rate of decay of the substance.?

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Answer 1 · 2018-04-21T23:59:36

#a'(t)=-100(ln1.1)(1.1)^-t#

#a'(t)#, the rate of decay, can be found by differentiating #a(t).#

#a(t)=100(1.1)^-t#

To make differentiation easier, rewrite using the fact that #x=e^lnx:#

#a(t)=100e^(ln[(1.1)^-t)]#

Recall that #ln(a^x)=xlna:#

#a(t)=100e^(-tln1.1)#

We can now differentiate this using the Chain Rule and the rules for differentiating the exponential function of base #e.#

#a'(t)=100e^(-tln1.1)*d/dt-tln1.1#

#a'(t)=-100(ln1.1)e^(-tln1.1)#

Recalling that #e^(-tln1.1)=e^(ln(1.1)^-t)=(1.1)^-t#, we get

#a'(t)=-100(ln1.1)(1.1)^-t#