Page:Calculus Made Easy.pdf/182

 It is found by putting this equation in its logarithmic form, namely,

which gives

(3) The quantity $$Q$$ of a radio-active substance which has not yet undergone transformation is known to be related to the initial quantity $$Q_0$$ of the substance by the relation $$Q = Q_0 \epsilon^{-\lambda t}$$, where $$\lambda$$ is a constant and $$t$$ the time in seconds elapsed since the transformation began.

For “Radium $$A$$,” if time is expressed in seconds, experiment shows that $$\lambda = 3.85 \times 10^{-3}$$. Find the time required for transforming half the substance. (This time is called the “mean life” of the substance.)

We have $$0.5 = \epsilon^{-0.00385t}$$.

Exercises XIII. (See page 260 for Answers.)

(1) Draw the curve $$y = b \epsilon^{-\frac{t}{T}}$$; where $$b=12$$, $$T=8$$, and $$t$$ is given various values from $$0$$ to $$20$$.

(2) If a hot body cools so that in $$24$$ minutes its excess of temperature has fallen to half the initial amount, deduce the time-constant, and find how long it will be in cooling down to $$1$$ per cent. of the original excess.